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HELICOPTERS - CALCULATION AND DESIGN
Volume 11. Vibrations and Dynamic Stability
by M. L. Mil% et al.
^^Mashinostroyeniye'^ Press
Moscow, 1967
.---.ad ■ '^. 'lA
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON;:d. C.^' ^ MAY 1968
TECH LIBRARY KAFB, NM
72
DDbB
NASATT F-519
HELICOPTERS - CALCULATION AND DESIGN
Volume II. Vibrations and Dynamic Stability
By M, L. Mil , et al.
Translation of "Vertolety. Raschet i proyektirovaniye.
2. Kolebaniya i dinamicheskaya prochnost\"
"Mashinostroyeniye" Press, Moscow, 1967.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
For sale by the Clearinghouse for Federal Scientific and Technical Information
Springfield, Virginia 22151 - CFSTI price $3.00
ANNOTATION [^^
The work "Helicopters, Calculation and Design" is pulDlished in three
vol-umes :
Vol.1 - Aerodynajnics;
Vol. II - Vibrations and Dynamic Strength;
Vol. Ill - Design.
The second volmie gives an account of certain problems of the theory of
vibrations and methods of calculating stresses set ip during such vibrations in
helicopters in flight, and, in particular, in the rotor blade.
Methods are presented for calculating the service life of a structure and
for calculating helicopter vibrations which permit determining the anplitudes
of these vibrations and conparing them with the norms of comfort. For the
first time in Soviet literature, the problem of co*L:pled vibrations of rotor and
fuselage is examined.
The theory of self-excited oscillations of a special type known as "ground
resonance" is discussed in detail. The characteristics of the occurrence of
such vibrations in a helicopter on the ground, during takeoff and landing run,
and under flight conditions are examined.
Special cases, little elucidated in the general literature, of calculating
bearings that operate under specific conditions of rolling are examined in a
separate chapter. The sajne chapter gives an account of the theory and method of
calculatljig a new type of thrust bearing of high load capacity and bearings
under com.pound loads.
The book is intended for engineers of design offices, scientific workers,
graduate students, and teachers of higher institutes of learning. It might be
useful to engineers of helicopter manufacturers and to students for furthering
their knowledge of the vibrations and dynamic strength of helicopters. Certain
sections of the book will be useful also to flight and technical staffs of heli-
copter flight units.
There are 35 tables, 2^6 illustrations, and 47 references.
Cand. Tech. Sci. R.A.Mikheyev, Reviewer
-- Numbers in the margin indicate pagination in the foreign text.
IX
PREFACE 12_
The first vol-ume of the work "Helicopters, Calculation and Design", pub-
lished in 1966, was devoted to aerodynamics: theory and methods of calculating
the aerodynamic characteristics of rotors and an aerodynamic calculation of
helicopters of various configurations.
That volume included an account of the theory of rotor flutter which usual-
ly belongs to the category of aeroelasticity - an area between aerodynamics and
mechanical strength.
The present, second volume is a logical continuation of the first and is
devoted to vibrations and dynamic strengths of helicopters •
The problems of the static strength of helicopters comprise no fundamental-
ly new aspects in comparison with what is known in aircraft construction. With
respect to vibrations and dynamic strength, helicopters exhibit a number of pe-
culiarities which were recognized when they first appeared as a new type of fly-
ing machine. These peculiarities loomed large during - if one may use the ex-
pression - the "struggle for existence" of this new type of craft in the overall
system of air transport means not requiring airfields.
The recital of the problems of vibration and dynamic strength of the heli-
copter begins with a description of a method of calculating the elastic vibra-
tions of its rotor blade, which are similar in fundamental equations and methods
of solution to those used in the theory of flutter but have a different trend
since ultimately the calculation reduces mainly to a solution of the purely me-
chanical strength problem, namely to a determination of variable stresses acting
in the blade, and then, with the use of data on the fatigue limits of a specific
structxire, to a determination of service life, i.e., blade life.
Problems of vibrations and dynamic strength are inportant not only from the
viewpoint of reliability of the craft. Also the service life of machines, and
hence their economy, depends on the solution of these problems.
In particular, this volume examines current methods of calculating elastic
vibrations of a blade, performed on high-speed electronic conputers which per-
mits determining the variable stresses set ip in the blade.
Investigations of the "ground resonance" mode of vibration, just as a study
of the vibrations of a structure, constitute the principal theme of the theory
of helicopter vibrations.
Elimination of "groiind resonance" vibrations which, if they arise and de-
velop further, lead to destruction of the craft on the ground and, in the case
of multirotor configurations, also in the air, has always been one of the main
problems confronting the designer. The problem of vibrations of helicopter
parts, examined from the viewpoint of crew and passenger comfort, is also quite
inportant. It is not difficult to estimate the acuteness of this problem when
iii
thinking of the power of the constant source of such vibrations - a huge rotor
operating in a highly variable velocity field. /h
The last chapter of this volume is devoted to a calculation of special
bearings, a necessity in designing many of the helicopter conponents and thus
representing a transitional chapter to the third volume on "Helicopter Design".
The volume "Design" will give a brief study of the main problems in layout
of helicopters, selection of the basic parameters of helicopters including
winged types, and auxiliary propulsion units such as tractor propellers or stp-
plementary jet engines. Economic considerations of aviation engineering, of im-
portance in designing, will also be presented.
This voliome also presents a discussion of problems of balancing, controlla-
bility, and stability fvom the viewpoint of selecting parameters for the control
system, as well as problems of designing individual conponents of the helicopter.
M.Mil»
The second volume "Vibrations and Dynamic Strength" was written by:
Introduction, M.L.Mil»; Chapter I, A.V.Nekrasov, Chapters II and III, L.N.
Grodko; Chapter IV, M.A.Ieykand. Section U of Chapter I was written by A.V.
Nekrasov in collaboration with engineer Z.Ye.Shnurov.
In preparing the manuscript the authors were assisted by engineers F.L.
Zarzhevskaya, V.M.Kostromin, and I.V.Kurov.
In this volume, we made use of the results of calculations performed by
engineers Yu.A.Myagkov, O.P.Bakhov, V.F.Khvostov, S. A. Go lubtsov, V.M.Pchelkin,
S.Ye.Sno, V.G-.Pashkin, N.F.Shevnyakova, N.M.Kiseleva, L.V.Artamonova, V.F.Semina,
N.A.Matskevich, V.I.Kiryushkina, and A.G.Orlova.
The reviewer, R.A.Mikheyev, offered many valuable comments.
Engineer L.G.Rudnitskiy was in charge of the final preparation of the manu-
script for publication.
The authors e^q^ress their sincere gratitude to these coworkers.
IV
TABIS OF OONTMTS
Preface •«««••••••• ,
Introduction , . , .
CHAPTER I ELASTIC VIBRATIONS AND BLADE STRENGTH
Section !• ProlDlems of Calculation, Basic Assunptions, and
Derivation of Differential Equations of Blade
Bending Deformations .••*♦. #.•..
1. Ultimate Purpose of Calculating Elastic
Blade Vibrations
2. Calculation of Blade Strength
3« Flight Regimes Detrimental to the Fatigue
Strength of the Structure
4« Assumption of a Uniform Induced Velocity Field
5* Assuirptions in Calculating Aerodynamic Loads
on the Blade Profile
6. Relation of Deformations due to Bending in
Two Mutually Perpendicular Directions and
Corresponding Assurrptions for Calculation ..•.
?• Consideration of Torsional Deformation of a
Blade in Calculations of Flexural Vibrations
S. Two Calculation Steps in Blade Design:
Calculation of Natiiral Vibration Frequency
and Calculation of Stresses
^• Idealized Blade Models Used in Calculation ....
10. Derivation of the Differential Equation of
Blade Bending in a Centrifugal Force Field
at Vibrations in the Flapping Plane
11. Differential Equation of Blade Bending in the
Rotor Plane of Rotation ,
Section 2. Free Vibrations of the Blade of a Nonrotating
Rotor
1. Method of Calculation for Solution of the
Integral Equation of Blade Vibrations
2. Calculation of the Natural Vibration Modes and
Frequencies of a Blade Model -with Discretely
Distributed Parameters . # # • . .
3. Condition of Orthogonality and Calculation of
Successive Natural Vibration Harmonics
4» Characteristics of Calculation of Natural
Vibration Frequencies and Modes of a Hinged
Blade
5. Calculation of the Natural Vibration Modes and
Frequencies of a Blade as a Siir^^ly Sipported
Beam •
Section 3- Approximate Method of Determining the Natural Blade
Page
iii
1
11
12
12
13
14
15
15
16
17
la
18
20
22
22
22
24
26
27
28
mill II I iiiinri
Page
VilDration Frequencies in a Centrifugal Force
Field
1. Use of B.G.Galerkin's Method for Determining
the Natural Blade Vibration Frequencies
2» Resonance Diagram of Blade Vibrations
3* Selection of Blade Parameters to Eliminate
Resonance during Vibration in the Flapping
Plane • • .
4* Selection of Blade Parameters to Eliminate
Resonances in the Plane of Rotation
Section 4* Calculation of Natural Blade Vibration Modes
and Frequencies in a Centrifugal Force Field
1. Purpose and Problems of Calculation
2. limits of Applicability of Calculation
Methods Reducing to a Solution of the
Integral Equation of Blade Vibrations
3. Possible Methods of Calculating Free Blade
Vibrations in a Centrif -ugal Force Pleld
4. Three-Moment Method for Calculating Natural
Blade Vibration Modes and Frequencies in a
Centrifugal Force Field
5. Determination of Bending Moments on the
Basis of Knovm Forces
6. Determination of Displacements from Knovm
Bending Moments
7. Case of a Blade Rigidly Attached at the Root
8. Possible Sinplifications in Calculating
the Coefficients
9. Certain Results of Calculating the Natural
Blade Vibration Modes and Frequencies
Section 5. Torsional Vibrations of a Blade
1. Problems Solved in Calculating Torsional
Vibrations
2. Differential Equation of Torsional Blade
Vibrations
3. Determination of the Natural Torsional
Blade Vibration Modes and Frequencies
4« DetexTiiination of the Natural Vibration Modes
and Frequencies of a Rotor as a Whole
Section 6. Combined Flexural and Torsional Blade Vibrations ..
1. Coupling of Flexural and Torsional Vibrations
2. Method of Calculating Binary Vibrations
3. Effect of Coupling between Bending and
Torsion at Natural Vibration Frequency ...#....
Section 7 . Forced Blade Vibrations
1. Use of B.G.Galerkin's Method for Calculating
Blade Deformations. Determination of Static
Deformations of a Blade
2. Determination of Blade Deformations with Periodic
implication of an External load
29
29
31
32
34
36
36
37
39
39
46
48
50
50
51
57
57
58
60
64
65
65
66
71
77
77
79
n.
Page
3» Sinplified Approach to Calculation of
Forced Blade Vilbrations •• *
4- Anplitude Diagram of Blade Vibrations
5. Calculation of Vilbrations at implication
Phase of External load Variable over the
Blade Length *
6. Aerodynamic Load on a Rigid Blade .*
?• Determination of the Blade Flapping
Coefficients *
8. Simplified Calculation of Elastic Blade
Vibrations •
Section S. Calculation of Bending Stresses in a Blade
at Low and Moderate Flying S|peeds
1. Characteristics Distinguishing Flying
Regimes at Low and Moderate S|peeds
2. Method of Calculating Stresses
3» Assunptions in Determining Induced
Velocities •.«
4« Mathematical Formulas for Induced
Velocity Field Determination
5 • Transformations of Mathematical Formulas
in Particular Cases .-
60 Nimerical Determination of the Integrals
J(PJ and J(Pj
7. Assunptions Adopted in Aerodynamic
Force Determinations • •...•..
8 . Mathematical Formulas
9 • Conversion to an Equivalent Rotor • «
ID. Basic Assunptions Used in Calculation
of Bending Stresses *
11. Differential Equation of Blade Vibrations
and its Solution
12 • Determination of the Coefficients on the
Left-Hand Side of the Equations in Table 1.8
13. Determination of the Coefficients on the
Right-Hand Side of the Equation of
Table 1.8
14 • System of Equations after Substitution of
Eqs.(8.34) and (8.38)
15. General Computational Scheme
16. Determination of Deformation Coefficients
17 . Computational Program •...
18. Conparison of Calculation with E^^eriment
at Low Flying S^eed
19. Comparison of Calculation with Experiment
at Moderate-S|peed Mode
20. Possibilities of Further Refinement
of Calculation Results • •
Section 9. Calculation of Blade Bending Stresses, with
Consideration of the Nonlinear Dependence of
81
83
84
S5
89
93
97
97
97
98
100
102
103
107
107
111
111
112
113
116
117
IIB
118
120
122
12^,.
125
vai
Page
Aerodymanri c Coefficients on Profile Angle
of Attack and Mach NijmTDer 127
1. Flight Regimes 127
2. Deterniination of Aerodynamic loads ••• 127
3. Method of Blade Calculation as a Siystem
whose Motion is Coupled "by Prescribed
Vibration Modes 129
4* Mathematical Formulas for a Blade Model
with Discrete Parameters 133
5« Consideration of a Variable Induced
Velocity Field 134
6. Characteristics of Numerical integration
of Differential Equations of Elastic
Blade Vibrations ••••* 135
7. Numerical Integration Method Proposed by
L.N.arodko and O.P.Bakhov 143
8. Sequence of Operations in Recalculation
and Practical Evaluation of Different
Integration Steps •• •...• 144
^• Conparison of Results by Numerical
Integration Methods with Calculation
of Harmonics 147
10 • Seme Calculation Results 148
Section 10. Calculation of Flexural Vibrations with Direct
Determination of the Paths of Motion of Points
of the Blade ^ 151
1. Principle of the Method of Calculation 151
2* Determination of Elastic Forces implied
to a Point of the Blade by Adjacent
Segments 153
3» Characteristics of Ntanerical Integration
of Eqs . ( 10 . 1) 156
4. Equations of Motion for a Multihinge
Articulated Blade Model 158
5. Sequence of Operations in Calculating
Elastic Vibrations by the Numerical
Integration Method 161
6. Method of Calculation with Inverse Order
of Determining Variables in Numerical
Integration I63
7. Conparative Evaluation of Various Methods
of Calculating Flexural Blade Vibrations I65
Section II. Fatigue Strength and Blade life 168
1. Testing a Structure to Determine its
Service life 168
2. Dispersion of the Characteristics of
Endurance in Fatigue Tests I69
3- Basic Characteristics of the Fatigue
Strength of Structure 170
4- Stresses Set U|5 in the Blade Structure
in FUght 173
viix
Page
5. Ifypothesis of linear Summation of Damage
Potential and Average Equivalent MpHtude
of Alternating Stresses • . . . •
6. Dispersion of the An5)litudes of Alternating
Stresses in an Assigned Flight PJegime •#•••
?• Method of Calculating Service life -with the
Use of Reliability Coefficients .*...
8. Method of A.F.SeHkhov for Calculating the
Required Safety Factor with Respect to the
Number of Qycles T|n •
9. Determination of Siogu at Given Fiducial
Probability ....•.♦.
10. Dispersion in the Stress Levels for Various
Struct\iral Specimens and Reliability Margin
with Respect to the Airplitude of Alternating
Stresses Tl(j • •
11. Method of Deteniiining the Reliability
Margin T|o- Proposed loj A#F.Selikhov
12. Exarrple of Calculation of Service life ....
13. Possible Ways of Determining the Minimum
Endurance Limit of a Structure
lU* Advantages and Disadvantages of Various
i^proaches in Determining the Necessary
Reliability Margins, and Estimation of
their Accioracy
15. Blade Strength Requirements in Design
Selection
16. Strength of a Blade with Tubular Steel
Spar
17. Strength of a Blade with Duralumin Spar ...
IB. Effect of Service Conditions on Fatigue
Strength of Spars
CHAPTER II HELICOPTER VIBRATIONS
Section 1. Forces Causing Helicopter Vibrations
1. Excitation Frequencies
2. Dependence of the Frequency Spectrum of
Exciting Forces on the Harmonic Content
of Blade Vibrations -
Section 2* Flexural Vibrations of the Fuselage as an
Elastic Beam
1. Calculation of Forced Vibrations of an
Elastic Beam by the Method of Expansion in
Natural Modes
2. Dsmamic Rigidity of a Beam. Resonance and
Antiresonance
3. Application of the Method of Dynamic Rigidity
to the Vibration Analysis of Side-by-Side
Helicopters
175
179
180
1B3
187
189
194
198
202
205
206
207
211
212
21U
21U
21U
217
227
228
234
237
XX
Page
4. Method of Aiixiliary Mass 2^1
5. Effect of Danping Forces. Vibrations at
Resonance • 2^2
Section 3. Vibration Analysis with Consideration of
Fuselage Characteristics 2^7
1. Fuselage Characteristics. Lateral and
Vertical Vibrations 2kl
2. Calculation of Fuselage Vibrations in the
Plane of Symmetry by the Method of Residues . . . 251
3. Consideration of the Effect of Shearing
Deformation 258
Section 4» Combined Vibrations of the System Fuselage- Rotor .. 259
1. Vibrations of the System Fuselage-Rotor 259
2. Calculation of the Natural Rotor Blade
Vibrations in the Plane of Rotation, with
Consideration of Elasticity of the Rotor
Shaft and Attachment to the Fuselage 263
CHAPTER III GROUND RESONAl^CE 274
Section 1. Stability of Rotor on an Elastic Base 275
1. Statement of Problem and Equations of
Motion 275
2. Stability Analysis and Basic Results 281
3. Physical Picttire of Rotor Behavior in the
Presence of Ground Resonance 292
4. Rotor on an Isotropic Elastic Base 298
Section 2. lateral Vibrations of a Single-Rotor Helicopter .. 299
1. Preliminary Comments 299
2. lateral and Angular Stiffness of landing Gear.
Flexural Center 300
3. Natural lateral Vibrations of a Helicopter .... 304
4 . Determination of Danping Coefficients 308
5. Combined Action of the System Shock
Strut-Pneumatic Tire 310
6. Reduction of the Problem to Calculation of a
Rotor on an Elastic Base 312
7. Analysis of the Results of Ground Resonance
Calculations 314
Section 3* Characteristics of Danping of landing Gear and
Blade . Influence on Ground Resonance 315
1. Determination of the Danping Coefficient
of the Landing Gear Shock Absorber 315
2. Effect of Locking of the Shock Absorber as a
Consequence of Frictional Resistance of the
Gland and Self-Excited Vibrations of the
Helicopter 318
3. Characteristics of Blade Danpers and
their Analysis 322
Page
4» Effect of Flapping Motion of Rotor on
Ground Resonance #
Section 4* Ground Resonance of a Helicopter during
Ground Run
1. Stiffness and Danping of a Wob^bling Tire
2» Calculation of Ground Resonance and Results
3# Ground Resonance on Breaking Contact of the
Tires with the Ground
Section 5* Ground Resonance of Helicopters of Other
Configurations
1# General Comments
2. Calculation of lateral Natural Vibrations
■with Consideration of Three Degrees of
Freedom •••••• •
3* Calculation of Natioral Helicopter Vibrations
in the Plane of Symmetry (Longitudinal
Vibrations)
4* Reduction of the Problem to Calculation of a
Rotor on an Elastic Base
5. Self- Excited Vibrations in Flight of a
Helicopter -with an Elastic Fioselage
Section 6 . Selection of Basic Parameters of Landing Gear
and Blade Danpers. Design Recommendations •«.
1. Selection of Blade Dairper Characteristics
2. Rotor -with Interblade Elastic Elements
and Daupers
3. Selection of Stiffness and Danping
Characteristics for landing Gears .•••
4. Certain Recommendations for landing
Gear Design
CHAPTER IV THBORETICAL PRINCIPIES OF CALCULATING BEARINGS
OF MAIN HELICOPTER COMPONENTS
325
330
330
335
337
339
339
340
347
352
354
356
357
361
364
367
369
Section 1. Equations of Static Equilibrium of Radial and
Radial- Thrust Ball Bearings under Combined Load
Section 2* Calculation of Radial and Radial-Thrust Ball
Bearings under Combined loads, for Absence of
Misalignment of the Races . . . . •
1# Pressiore on Balls
2. Reduced Loads • <
3. Statistical Theory of Dynamic load-Carrying
Capacity *
4. Effect of Axial load on Bearing Performance
5. Approximate Solutions of Equations (2-1)
and (2.2) <
6. Relative Displacements of Races
Section 3# Certain Problems in Calculating Radial-Thrust
Ball Bearings -with Consideration of Misalignment
of their Races under Load *
370
377
377
384
385
393
397
403
405
XI
Page
1* Basic Relationships
2. Case of "Piire" Moment •••••••
3» Simultaneous Action of Moment and Axial
Force
4 • limit Dependences on Sbiall loads •••••••••«••
5. Distrit)ution of Load iDetween Rows of Balls
of Double-Row Radial-Thrust Ball Bearirigs
6* Exanples of Calculation ••#
Section 4» Calculation of Tapered Roller Bearings under
Combined Loads ••
1. Calculation of Single- Row Tapered Roller
Bearings • •
2» Remarks on Calculation of Bearing Assemblies
of Two Tapered Roller Bearings
Section 5. Calculation of Vibrating Bearings <
1. Characteristics of the Mechanism of Wear of
Antifriction Bearings under Vibration
Conditions •• • .••#.. <
2* Lubrication of Highly Loaded Vibrating
Bearings in the Presence of Small Vibration
AnpHtudes -
3* Calculation of Hub Bearings in Main and Tail
Rotors ••• ••• ..-..#.
4. Calculation of Bearings for the Pitch Control
and Control Mechanisms •• -
Section 6. Theory and Selection of Basic Parajneters of Thrust
Bearings with "Slewed" Rollers • • -
1. Determination of the Time T^
2. Selection of Angles of Slope of Cage Seats
3 . Friction Losses • • . . •
k* Additional Considerations of Optimal Thrust
Bearing Design with "Slewed" Rollers
5. Exanple of Calculating a Thrust Bearing
with "Slewed" Rollers
References •
405
409
413
4-5
421
424
426
426
431
433
434
435
kkl
450
453
454
459
463
465
466
468
xzi
INTRDDUCTION /^
As soon as an aircraft engine of sufficient power and light in weight had
been created and the first helicopter took off from the ground, prolblems of
balancing, controllability, and stability of this craft arose. These were main-
ly aerodynamic problems • If we consider the first flight of de la Cierva^s
autogiros in 1925-19^6 to be the start of flight of rotary-wing aircraft, then
we can say that the stated problems were mainly solved in the first decade
(I926-I936) of their development. The new type of flying machine was thus cured
of its "childhood diseases".
However, as soon as the first series-produced machines appeared and they
were placed in service, more serious deficiencies of helicopters became apparent
such as, for exanple, fatigue due to insufficient dynamic strength of certain
structural members .
New dynamic problems arose with the wider practical use of autogiros and
especially of helicopters, which entered the scene at the end of the Thirties
and beginning of the Forties on a new inproved technical basis. These problems
pertained prijnarily to oscillations and vibrations of individual structural ele-
ments and of the helicopter as a whole, which are harmful owing to the stresses
set ip in this case, or are inpermissible from the viewpoint of necessary crew
and passenger comfort, and also include the problem of service life of struc-
tural elements operating under high variable stresses. The latter problem,
namely the increase in service life, is constantly gaining in importance
since the amortization and overall life of a helicopter, determined by the life-
times of its conponents, has an effect on the cost-effectiveness in its use as
a means of transportation. The service life, in turn, is determined mainly by
the level of the variable stresses set up in the structiore; therefore, the accu-
racy with which these are calculated is one of the basic problems of a dynamic
strength analysis of helicopters.
A tractor propeller of a conventional aircraft operates practically in an
axial flow and, like an engine, sets vp no noticeable variable stresses in the
structural members. Only takeoff, landing, and flight under conditions of atmo-
spheric turbulence (and, on military aircraft, maneuvers) create appreciable
dynamic loads on the aircraft structure, but at relatively few load cycles (of
the order of tens and hundreds of thotisands of cycles) during the lifetime of
the aircraft. In this case, one can speak about repeated static loads.
The loads on the helicopter are quite different. Its main structural
members are loaded dynamically, the nimaber of loadings often exceeding tens of
millions of cycles duiang its lifetime. This is due primarily to the asymmetric
flow past the rotor, which rotates and simultaneously advances. In so doing, /6
the blade is subject to variable aerodynamic loads as a consequence of the change
in relative flow velocity and angles of attack of its sections. All forces and
moments acting on the blade are transmitted to the hub and rotor controls. The
forces and moments arriving from different blades are mutually conpensated, with
the exception of loads acting with frequencies whose ratio to the rotor rpm is
"riiiiii
illlllliillllllllll
a multiple of the "blade n-um"ber. These loads are transmitted to the fuselage
and to the nonrotating part of the rotor control system and there set up notice-
able variable stresses •
Thus, the problem of vibrations and dynamic strengths in helicopter conr-
struction is not only much broader than in aircraft construction but, in many
cases, has no direct analogy in the latter.
Recognition of the nnportance of the problems of dynamic strength was not
immediately obvious. Thus, even the causes of the first accidents of autogiros
in I936-I937, during which these craft overttirned in the air, were attributed
for long to insufficient dynamic stability. In this respect, in particular, in-
vestigations of the dynamics of a rotor with hinged blades at curvilinear motion
of the craft were undei*taken (see Sect. 2, Chapt.II of Vol.1). This theory later
found wide application in the elaboration of problems of dynamic stability and
control "lability of helicopters. However, it never uncovered the true cause of
the above-mentioned accidents. As was subsequently realized, the cause was in-
sufficient dynamic strength of the rotor blades .
These problems were recognized literally by hit and miss. The first experi-
mental autogiros and helicopters were small and thus had a rather high structural
rigidity. However, the first increase in size immediately encountered consider-
able difficulties. For instance, on the A-4 aiibogiro, which had a diameter
somewhat larger than its predecessor the 2EA autogiro, serious difficulties arose
owing to insufficient torsional rigidity of the blade. The blade angle, in the
first flight, increased so much due to torsional deformation that autorotation
was inpossible and the flight almost ended in crackoj^D.
The investigation of this phenomenon was conpleted with the publication of
a paper on the dynamic twisting of a rotor blade in flight [see (Ref .2)], in
which the first suggestions were made as to the necessity of matching the center
of gravity and the center of pressure, and in which considerations of the effect
of blade profile on static stability and controllability of the craft were ex-
amined. This investigation resulted in asymmetric profiles, ensuring a large
reserve of autorotation, which were adopted in the engineering practice of Soviet
helicopter construction. A set of different profiles was used for the blade
arrangement. The recommendations in the above paper were stiff icient to prevent
flutter in the first Soviet helicopters which had a rotor span of about 14 ni.
The development of the Soviet helicopter industry is characterized by
larger steps than that of the helicopter industry in other countries (this also
enabled Soviet designers, who had started later to build helicopters, to create
machines vastly superior to modern foreign helicopters with respect to lift
capacity and size). Whereas, after the first successful flight of the Sikorsky
S-51 with a rotor span of 14 m built in 1947, the Americans, in I95O-I95I, began
working on a craft with a rotor of I5.5 ni diameter (3-55), the Soviet designers,
after creating the Mi-1 helicopter with a 14^m rotor, constructed as early as
1952 the Mi-4 and Yak- 24 helicopters with 2l-m rotors. It is not surprising
that such a jimp in size led to a new previously unencountered phenomenon: In
both craft, the rotor began to flutter during the first takeoff. We readily /2
coped with this problem, but problems of the theory of flutter had to wait a
long time for solution.
We first encoiontered this new phenomenon in April 1952 with the Mi-4 heli-
copter, when it was ready for its maiden takeoff • At onset of overspeeding,
the "blades "began to flap in a random manner, "bending to an ever increasing ex-
tent and threatening to strike the airframe- The test crew realized that this
was a new phenomenon never "before encountered- This constituted so-called
flutter of the rotor blades. At that time, no one thought of the fact that this
was the very same type of flutter under study ^j many scientists in the USSR and
other countries. According to the data availalDle at that time, flutter was not
expected since it was thought to arise at a"bout 500 rpm rather than at a rotor
rpm of 100-110, as actually happened in the Mi-4 helicopter. The decisive
factor for the occurrence of flutter in this case was the fact that the large
forces generated on a rotor of such a diameter produced appreciable deformation
of the swaslplate of the automatic pitch control, which is equivalent to a de-
crease in torsional rigidity of the "blades, and also the fact that a large
value of the coefficient of the flapping conpensator (close to unity) had been
selected for these machines; this point had been disregarded in earlier investi-
gations of flutter* As a result there was no reason to think of helicopter
flights, since flutter set in appreciable before the operating rpm of the rotor
was reached.
It became clear, in studying the pattern of flutter (flapping, bending, and
twisting of the blades) that this phenomenon could be prevented only by utiliz-
ing the torques from the inertia forces generated during displacement of the
blade sections on flapping. Without associating rotor flutter with wing flutter
where - as known for long - the mutual position of center of gravity, f lexural
axis, and center of pressure is of prime inportance, we sinply attached counter-
weights to several points along the blade length to create moments of inertia
of opposite sign during vibrations and then repeated start-icp of the rotor; we
ijimediately understood that we had in hand a reliable means of stopping flutter.
Thus, within a short time this problem was practically solved, and by May
1952 the first fUghts with the Mi-4 helicopter were made.
At the same titne, flutter occurred also on the Yak-24 helicopter which had
the same hub and automatic pitch control as did the Mi-4 helicopter but the
blades were of different design (with larger flexural and torsional rigidity) .
However, as a consequence of the fact that the rigidity of the automatic pitch
control and the parameters of the flapping conpensator were decisive in the oc-
currence of flutter, we also encountered flutter of the very same form and at
the same rpm on the blades of the Yak-2iV helicopter as well as on the Mi-4 heli-
copter.
Thus, within several weeks a practical solution was found for preventing
flutter, which is used even now. The scientific theoiy, however, to detemdne
whether flutter will or will not occur and - if it did occur - at what ipm and
in what form, was developed by us during the subsequent four years.
It should be stated that, after conpletion of studies of flutter on the
ground (by shifting the centering of the blade forward it was possible to "drive"
it beyond the limits of the working rpm and even beyond the maximum permissible
rpm of the engine on the ground), there still existed the possibility of its
occurrence in flight. This led to detrimental happenings. In January 1953>
crash of a Mi-4 helicopter took place, whose causes were not satisfactorily /8
defined for almost three years. Inspection revealed traces of impact of the
blades on the cockpit. This had never been observed before. ¥e should note
that, dioring normal flapping motion, the blade does not come into contact with
the cabin unless the lower restrictors of blade overhang or coning stops are
rtptured in the air.
It is obvi-ous that our search for the cause of this accident was diligent,
when realizing that the crash did not stop either actual flights or series pro-
duction of this prototype.
During 1954* a number of pilots observed an unusual phenomenon in flight,
which came to be known as "Kalibernyy effect" (after the pilot who was the first
to notice it). Kalibernyy established that in a power descent at a blade set-
ting angle of about 6 - 7^, the blades began to flap out of their coning angle.
This stopped after resetting the blades that had a somewhat different transverse
centering. However, two years later, when flight-testing a set of blades for
absence of the "Kalibernyy effect", i.e., during a power descent with an angle
of pitch of 6 - 7^'''", this phenomenon became so predominant, at such strong flap-
ping of the blades, that it was difficult to make a forced landing with the
craft. It should be mentioned here that close to the ground, iipon changing to
another regime, the blade flapping stopped and the craft behaved normally. A
visual inspection of the helicopter after the flight revealed ruptured blade
footings (so-called movable slotted trailing edge of the blade), which indicated
bending of the blade in the plane of rotation. Everything else was in good
working order. It was decided to make a detailed checkout of this helicopter
with the same set of blades. Flight tests were carried out to check and study
this phenomenon.
Measurements of the blades showed that the centering had shifted by about
1% of the chord more rearward than its position at the time of the blades leav-
ing the manufacturer. This can be explained as follows: The blades were
sheathed with plywood. The center of gravity of plywood is about at 50% of the
chord. Therefore, as soon as the wood swells and increases in weight due to the
absorbed moisture, the center of gravity of the entire blade will shift toward
the trailing edge. The above happening with the helicopter occurred in autumn
when the humidity was high.
During these tests it was also conclusively established that the character
of blade flapping and the motions of the control stick during flight in a
"Kalibernyy effect" regime are conpletely analogous to the flapping and motion
of the stick recorded in ground tests where blades are caused to flutter by an
artificially created tail- heaviness. This coup lex procedure made it possible to
establish that the phenomenon occurring in flight was identical with that noted
on the ground. Thus it was established that the "Kalibernyy effect" is none
other than the onset of flutter in flight. On the basis of this conclusion it
was conjectured that the earlier iinexplained flight accident in which the blades
struck the cockpit was also nothing else than flutter of the blades in flight
arising at a rotor rpm at which it did not appear when operating on the ground.
-5^ On the Mi-4 helicopter, flutter sets in primarily in this regime.
4
The vibrations of a hinged blade in flutter, unlike the vibrations of a
conventional aircraft wing, are capable of a flapping motion whose airplitude
builds up until the blade inpacts on the coning stops and, after breaking these,
strikes the cabin •
That this phenomenon was not uncovered for a long time can be attributed /2.
to the erroneous assuiiption based on model tests that, if flutter on the ground
is eliminated, it cannot ocovir in air during forward motion. However, practical
experience and, later, more rigorous experiments with helicopters and, finally,
corresponding theories have shovm that there are flight regijnes in which flutter
at the operating rpm of the rotor will not occur on the ground but may occur in
f night •
It should be stated that, as established in investigations, the phenomenon
of flutter also had been observed earlier on helicopters. Already in 1949, the
Mi-1 helicopter was equipped with a rotor with wider blades to increase the
safety factor relative to flow separation. In flight, this rotor produced buf-
feting which could not be eliminated. After the theory of flutter had been
worked out and all aspects of this phenomenon had been clarified, it became pos-
sible not only to attribute the jolting of the Mi-1 helicopter with wide blades
to an approach of the regime to flutter but also, and without further difficulty,
to design and construct (in 1956) a 35-meter rotor for the Mi-6 and Mi- 10 heli-
copters. Perfection of this rotor was confirmed by the fact that a week after
the initial takeoff the new heavy Mi-6 helicopter was able to corplete the train-
ing flight for participation in the Air Parade on Aviation Day at Tushino.
Neither then nor later did anything detrimental, associated with flutter, occvir
with these craft. This constitutes the historical aspect of the flutter problem.
Of no less ijirportance is the problem of determining variable stresses in
blades, which is solved by studying their forced vibrations.
During the first decade of their development, helicopter rotors were actual-
ly designed without precalculation of variable stresses arising in flight. At
that time, calculation was cumbersome and inaccurate and often conpleted only
after the craft was at the airfield. It was only the development of conputa-
tional methods for variable stresses, allowing the use of high-speed digital
coirputers, that permitted designing blades with deliberate selection of rigidity
and mass distribution so as to avoid harmful resonance, reduce the stress level,
and thus ensure long service life and blade reliability.
It should be noted that refinement of stress analysis for blades led to
further development in depth and elaboration of the aerodynamic theory. As
shown in the first volimie, refinement of the calculation of flight data did not
make it necessary to develop the conplex and cumbersome vortex theory of a rotor.
Nevertheless, it is only the vortex theory that permits determining the nonuni-
formity of the induced velocity field, causing variable blade loading at fre-
quencies that excite flexural vibrations of the blades of second, third, and
higher harmonics. Therefore, in stress analysis, only the vortex theory can
give results close to those observed in reality.
Vibrations constituted another no less important problem. This problem has
always been one of the most difficult in the development of rotary wing aircraft.
Dozens of Soviet and foreign designs, interesting from the viewpoint of concep-
tion and flight data, never came to conpletion owing to the high level of vi-
bration.
In conventional aircraft, the soiirces of vibration are not as powerful as
in helicopters. Fiirthermore, Tx)th engines and propellers which are the main
vibration exciters in conventional aircraft can be adequately isolated from the
structure by means of special shock absorbers. High-frequency resonance /lO
produced by such exciters can be eliminated quite easily by conparatively minor
modifications of the structure. In a helicopter, in addition to the fact that
the perturbing forces produced by the rotors are appreciably greater than in a
conventional aircraft, the frequencies from the slowly rotating rotor are rather
low and, in combination with the natural oscillation frequencies of the fuselage,
engine, wing, or tail unit, give rise to resonance leading to appreciable vibra-
tions with an amplitude of displacement which, in steady flight regimes, reaches
magnitudes of the order of 0.3 - 0.4 inm and in short-time regimes, prior to
landing of the helicopter, even 1 - 2 mm in the crew cabin.
Resonance with fundamental tones of the natural fuselage vibrations often
are practically inpossible to danp out by changing the rigidity of the structiire
in an already built machine. Therefore, it is iriqDortant to make a correct esti-
mate of the natijral vibration frequency of the fuselage and to calculate the
vibration amplitude in designing the craft.
In overcoming vibrations, main enphasis must be on reducing the magnitudes
of variable forces produced by the rotor and acting on the fuselage. These
forces are caused by blade vibration. In turn, such blade vibrations may be
larger or smaller depending on the closeness of their natural frequencies to the
frequencies of the external excitation sources.
In all cases, closeness to resonance will lead to an increase in blade
stresses. However, if the vibrations occur with the harmonic frequency z^, + 1
(or Zt3 - 1 for vibrations in the plane of rotation of the rotor) or with the
harmonic frequency z^ for vibrations in the flapping plane (where z^, is the
number of blades), then the forces are summed and transmitted over the hinges
to the hub and through it to the fuselage, causing vibration.
Vertical vibrations, which are the type most disagreeably perceived by man,
are largely caused by forces acting in the plane of rotation of the rotor, since
these forces, applied high above the center of gravity of the helicopter, create
appreciable moments that excite flexural vibrations of the fuselage. In this
case, it is natural that the greatest vibration anplitudes (antinodes) are
reached at the ends of the fuselage and hence in the cockpit.
It was found that, in determining the natural vibration frequencies of
helicopter blades, it must be considered that the rotor hub does not remain
fixed during the vibration since it is attached to an elastic fuselage. Thus,
in an analysis of vibrations, the craft should be treated as a single dynamic
system with elastic blades hinged to a hub attached to an elastic fuselage.
It is obvious that it is only lately that such a calculation scheme could
be developed and made available for study. As far as we know, we are the first.
in this book, to present a method of calculating helicopter vibrations in the
design stage ♦
later in this volume, we vdU discuss self-excited oscillations of a heli-
copter, generally known as "ground resonance".
Designers first encountered the phenomenon of ground resonance more than
30 years ago when one of the first Soviet autogiros, the A-6 (designed by V.A.
Kuznetsov), was equipped with low-pressure tires which were new at that time*
The oleo struts were removed from this helicopter • An unexpected vibration oc-
curred in the first takeoff attenpt. The helicopter rocked from wheel to wheel
at constantly increasing anplitude, finally junping i:pward so that the wheels
broke contact with the groiHid. The takeoff ended in failure.
Since the tests were recorded by a motion-picture camera, it was possible /1_1
to establish that the blades had executed increasingly stronger vibrations about
the drag hinge. These vibrations, which occurred in a centrifugal force field,
produced a periodic displacement of the center of gravity of the entire lifting
system relative to the center of the hub and thus excited vibrations of the heli-
copter standing on the ground. It is obvious that, if the frequency of dis-
placements of the rotor center of gravity coincides with the frequency of natu-
ral vibrations of the helicopter on pneumatic tires, such vibrations are able
to increase. It would seem that the physical aspect of the phenomenon is clear.
The energy that fed these increasing vibrations was either the energy of the
engine turning the rotor or, with the engine cut out, the kinetic energy of the
rotating rotor.
However, numerous investigations, which are still in progress, were needed
to develop the theory of ground resonance and to study its new manifestations,
possibly in new basically differing configurations and structures.
The first theoretical work e:xplaining the nat\H*e of self-oscillations of
the "ground resonance" type was done as early as 1936 by I.P.Bratukhin and B.Ya.
Zherebtsov. In particular, the results of their investigations made it possible
to eliminate ground resonance in the world^s largest autogiro, the A-I5 with a
rotor span of 18 m which was constructed in 1936 from the design by V.A. Kuznetsov
and M.L.Mil'. in the design of the hub of this autogiro, springs mounted to the
blade- vibration restrictor around the drag hinge were used. The springs were
given the natiu?al vibration frequency of the blades in the plane of rotation,
which eliminated "ground resonance".
There is no doubt that, at the time, the phenomenon of ground resonance was
also known in the Western Countries and had undergone some study there, since
even the first successfiil de la Cierva autogiros, for exarrple the C-19, had
elastic couplings (shock absorbers) connected to the blades over friction
danpers .
However, many designers continued for some time to produce autogiros with-
out danpers in the drag hinges. A model of such a machine was the A-7 autogiro
developed in 1937 by N.I.Kamov. It made successfiol flights without danpers on
the rotor hub. The secret of the success was the fact that this was the first
time a tricycle landing gear was used, which ensured a practically vertical
7
position of the rotor axis during engine re-\rving iDefore takeoff and after the
landing stop. This caused small initial perturbations due to deflection of the
"blades in the plane of rotation, since the initial deflections of the tilades are
produced tj the projection of the force of gravity onto the plane of rotation.
Another ijiportant point was the friction force in the hinges (at that time,
"bronze bushings were used in the hinges), which cannot "be disregarded in the
presence of appreciable centrifugal forces; in this case they produced suffi-
ciently large danping. On one occasion, the pilot S.A»Korzinshchikov after one
of the flights forgot to push the control stick immediately after landing and
thus did not change the craft from a three-point landing (tail skid and main
landing gear) to a standard position (with support on the front leg); ground
resonance occurred after subsequent decrease in rotor rpm owing to the large
initial disturbance in blade deflections in the plane of rotation (the axis of
rotor rotation was incHned at an angle of 14° to the ground), causing the
blades to break and the helicopter to be damaged.
Thus, the problem assimied constantly newer aspects from one experimental
model to another.
Since, at that time, no exact calculation of the required danping of blade
Aabrations existed (in the presence of ground resonance vibrations, the danping
of vibrations of the craft by shock absorbers on the landing gear is of equal /12
iirportance), designers atteirpted to select a minimum value of the friction
moment of the hub dairper. This was dictated by the desire to reduce variable
bending moments set vp in the presence of a darrper d-uring forced vibrations of
the blades in flight.
As is known, friction daupers cause vibrations at threshold excitation.
If the excitation is small, i.e., the excitatory moment is smaller than the
friction moment, no vibrations will appear. However, vibrations may suddenly
arise in a helicopter which is fail-proof with respect to ground resonance and
had already been in actual service* This can be attributed to the fact that,
in a given case, the initial pearburbations may be greater than usual. This case
occurred in the Mi-1 helicopter when taxiing obliquely across deep ruts made by
a truck. In this case, a random disturbance of tilt strongly rocked the craft
on its pneumatic tires, causing it to acquire such large vibration anpHtudes
that the available danping in the hub became inadequate and ground resonance
arose. The pilot G.A.Tinyakov remedied this in a sijnple manner by taking off;
this stopped the vibrations since the elastic coipling, i.e., the coipling with
the ground, was broken.
This case suggested the need for making use of visco-us friction, i.e., in-
stalling hydraulic blade vibration danpers in the hub, for which the moment of
friction does not remain constant but increases with the vibration anplitude.
However, practice constantly required inprovement and development of the
theory in this area. One merely need recall the generation of ground resonance
when the helicopter is attached to its moorings, with the engine operating.
Several cases of ground resonance were observed also when the wheels of the
helicopter, in taxiing during takeoff or landing, had only slight ground contact,
so that the propulsive force of the rotor came close to the weight of the craft
8
and the shock struts "with the usual pretightenLng were fully extended. The dif-
ference between the weight and the propulsive force of the craft was absorbed
only by the pneumatic tires.
It is obvious that, in this case, not only will the vibration frequencies
of the craft change but there also will be no danping of the struts. Thus,
ground resonance occurred here which had never ^::>een observed in a helicopter
that was not moored or was not taxiing, at very small wheel loading.
To avoid such cases, we began using so-called two-chamber landing-gear
struts, which were shock struts provided with a second low-pressure chamber for
absorbing the vibration energy of the craft when it made only slight ground con-
tact with the pneumatic tires while the main struts were not operative.
Problems of the theory of ground resonance are especially ojnrportant for
twin-rotor configurations when the elastic system coipling both rotors, be it
the fuselage in the fore-and-aft or the wing in the side-by-side configioration,
has low natural vibration frequencies. In the presence of such vibrations, ap-
preciable displacements of the rotor hub may take place, creating the possibility
of energy transfer between blade oscillations and oscillations of the Hfting
structure. Vibrations of this type are possible not only on the ground but also
in flight .
A similar problem arises in designing tail rotors with drag hinges mounted
to a flexible tail boom.
The development of harmonic and iirproved craft is possible only if the de-
signer is sufficiently con^^etent not only in general problems of design but also
in special problems having to do with the theory and calculation of the indi- /I3
vidual elements.
A modern helicopter contains many essential highly loaded mechanical com-
ponents whose reliability and service life depend in many respects on the per-
formance of the bearing assemblies* Consequently, helicopter designers should
be familiar with the theory and calculation of roller bearings. This pertains
specifically to cases of the work of roller bearings in coirplex combinations of
external loads and in the presence of rocking motion of low anplitude.
For this reason, we included a chapter giving answer to problems of the
theory and calculation of bearing assemblies of hubs, cyclic pitch control, and
other units. One of the most interesting problems described in Chapter IV is
the theory of special thrust roller bearings in which, owing to the positioning
of the rollers at an angle to the radial direction, the cage - during the rock-
ing motion - not only vibrates along with the movable collar but also continu-
ously rotates in one direction. This prevents local wear of the raceways and
increases the lifetime of the bearing.
It should be noted that the use of such bearings in the feathering hinge
of rotor hubs resulted in an appreciable increase in service life.
Helicopter engineering requires a high general level of theoretical and
scientific training of the design engineer, since dynamic problems are of much
greater ijoportance for helicopters (rotary wing aircraft) than for regialar air-
craft (prototypes with fixed wing, althotigh lately also including tilt wings and
variable sweep) . This is confirmed by the fact that the few designers who made
notable contributions to the development of helicopter engineering and especial-
ly those who had practical success, were simultaneously outstanding scientific
theorists. These include B.N.Yur*yev, Prof. A,M.Cheremukhin, and Prof. I.P*
Bratukhin who, in the Thirties, were the developers of the first Soviet heli-
copters from the lEA. to the llEA prototypes; Prof. Focke, the designer of the
FW-61 and FA-223 helicopters in Germany; one of the pioneers of aviation Louis
Breguet; Prof. Doran who created the first French helicopters; and many others.
It should be noted that the present level of theoretical training of de-
signers working for the foremost helicopter engineering firms of the world is
very high, as far as can be judged from the literature. For this reason, neither
the engineer-calculator nor the designer working in helicopter engineering should
have any difficulty in assimilating the material presented below.
The authors hope that this second volume will find readers and be foiind
usefiil.
The inserts show photographs of the main Soviet helicopters in series pro-
duction. These are the first Soviet series-produced helicopters with piston
engines Mi-1 and Mi-4, developed in 1949 and 1952. Having been produced in
large numbers, these prototypes range now among the most widespread variants of
helicopters .
Other photographs show the Mi-6 helicopter with two turboprop engines /1h
developed in 1957 and the Mi- 10 helicopter (I962) which is a flying crane with
a high landing gear, adapted for Hfting and transporting heavy stores rigidly
mounted on the underbelly. In I965, a world Hfting record for helicopters was
established with this cargo craft: 25 tons were lifted to a height of 2830 m.
The next pictures give the Mi-2 and Mi-8 helicopters which are a second
generation of Soviet light and medi-um versions. The lifting systems of the
Mi-1 and Mi-4 were retained on these, but the single piston engine was replaced
by two turboprop engines.
10
■ I II II I II MM 11
mill iiiiiiiiiiii II mil INI
CHAPTER I 715
ELASTIC VIBRATIONS AND BLADE STKEIMGTH
Calculation of elastic vibrations is a necessary element in the process of
developing new blade designs. It forms an inseparable part of the calculation
of blade strength.
To develop helicopter blades it is necessary to solve many presently quite
carrqDiex technological and design problems. In their solution, account must be
taken of the most diverse requirements and primarily of the requirement of high
fatigue strength of the structure.
The work of designing blades usually involves the following basic steps:
Selection of materials for individual structural members, determination
of optimal parameters, and design of the blade.
Selection of the best technological processes ensuring highest fatigue
strength of its main stressed elements, and manufacture of the blade.
Flight tests with analysis of stresses set up in flight.
Dynamic tests and evaluation of the blade service Hfe.
Performance of the coirplex of finalizing, including work on reduction
of active stresses and increase in fatigue strength of the structure.
Acceptance tests and start of series production.
Analysis of operation of series-produced blades under various high- load
and endurance conditions and layout of final designs for blade series
based on the analytical data.
Calculations of elastic blade vibrations are required at many stages of
this work, but primarily at the initial stage which terminates with the actual
blade design •
In selecting the blade parameters and its structural materials, one of the
main criteria is the magnitude of alternating stresses set vp in flight and the
correlation between these stresses and others characterizing the fatigue strength
of the structiore. It is only by calculations that the magnitude of these
stresses can be determined and an estimate made of the strength of the structure
at this stage. To design the blade within the required - usually rather short -
period, the designer should have available modern methods and conputational
means to obtain a rapid solution to any number of possible problems.
Of similar inportance is the calculation in the finalizing stage. As a
rule, in new blade designs the variable stresses are excessive, confronting the
designer with the problem of their reduction. For this, the occurrence pat- /16
tern of stresses measured in flight must be confirmed by calculation, followed
by devising means for their reduction by varying some of the parameters. To
attenpt a solution of this problem without calculation generally means excessive
loss of time in checking unverified asstmptions and waste of considerable funds
in manufacturing a blade that might be rejected after flight testing.
11
A reduction of alternating stresses is extremely iinportanb and. permits not
only an increase in the reHalDilLty and service life of the "blade "but also an
inprovement in mechanical and flying qualities of a helicopter such as, for
exanple, flying speed and Hft capacity, which in modern helicopters are often
limited "because of strength conditions.
Solution of all these problems would not be excessively difficult if the
calculation results would sufficiently well coincide with those observed during
in-flight stress analysis* Unfortunately, this is not always the case since
calculation does not necessarily give results satisfactory for practice.
Calculations for determining the natural vibration frequencies are most
reliable. Usually, an accuracy of the order of ±2% is achieved. Therefore, all
calculations on the exclusion of resonance yield high reliability. Galcular-
tions of alternating stresses at cruising and maximum flying speeds are notice-
ably less reliable. The stress values obtained in these calculations usually
are 15-25^ lower than stresses measured in flight. Consequently, the stress
analyses in these regimes do not always satisfy the designer. Nevertheless, the
error can be compensated to a certain extent by introducing into the calculation
a correction allowing for a constant divergence from e^q^eriment.
A still greater error is possible in calculations of alternating stresses
at low flying speeds.
It is obvious from the above that the calculation methods for alternating
blade stresses require further elaboration. Nevertheless, practice has shown
that parameter selection and blade finishing without even these imperfect methods
is rather ineffective. Therefore, this Chapter will give a detailed account of
various calculation methods. In ovir opinion, this will give the reader an idea
of all features of blade loading in flight, showing possible approaches for
calculation, for determining and estimating the advantages and shortcomings of
various methods and, finally, providing engineers concerned with such problems
bases for extension of studies and improvements of calculation methods.
Along with a description of various methods of calculating elastic blade
vibrations, on which main eirphasis is placed, this Chapter also presents the
basic principles of stress analysis for blades and of service life detexTidnation
(Sect. 11).
With respect to specific data on the selection of blade parameters, we
thought it preferable to include this problem in the Section "Blade Design"
forming part of the third vol-ume of this book.
Section 1. Problems of Calculation, Basic Assimiptions^ and
Derivation of Differential Equations of Blade
Bending Deformations
1. Ultmate Purpose of Calculating Elastic Blade Vibrations
The calculation of elastic blade vibrations is necessary in solving a
number of problems created in the designing and debugging of a helicopter. The
12
..■;.. ..v-. ^*;";!v^t*e::-j±fct.V^-
Mi-1 Helicopter •
ro
P3
s
Mi-4 Helicopter-
Mi-6 Helicopter •
H
o
'*■'#
'-^^''.\l ' ,ti^-:**'^U^?^
.^;^i^,^^^^■^^j'>iv^fi4^;--:W^4«4^
.:ft--'^--;MAv.-^.''^#f^
^K^.-^.^J|^^
Mi- ID Helicopter Crane •
^i-2 Helicopter.
.:::^..
Mi-8 Helicopter.
most inportant of these is the problem of determining alternating blade bending
stresses* Determination of these stresses forms the major part of the strength
calculation. Therefore, the main problem in this Chapter is to determine the /I?
elastic vibrations of a blade for calculating its strength.
A determination of blade vibrations is necessary also for solving many-
other problems. Without calculating these vibrations it is inpossible to deter-
mine the loads acting on the helicopter, the hub, its controls, and on the
transmission of the engine drive. A detennination of alternate loads exerted
on the helicopter by the rotor blades largely solves the problem of analyzjing
helicopter vibrations.
Also of interest is the problem of the effect of blade vibrations on the
handling qualities of the helicopter. The limitations inposed on the flying
qualities by flow separation due to the rotor blades are determined primarily
by the permissible anplitude of blade vibrations. With an increase in these
anplitudes, the variable forces in the controls and the vibrations of the heli-
copter increase. Therefore, a calculation of elastic blade vibrations permits
the most accurate estimate of the limits of helicopter flight regimes with re-
spect to flow-separation conditions.
To some extent, blade vibrations - and primarily torsional vibrations - af-
fect the aerodynamic characteristics of the rotor even when far removed from
regijnes "with flow separation.
We will discuss the first of the above problems in greater detail.
2. Calculation of Blade Strength
Calculation of blade strength involves a determination of the constant and
variable stresses at all points of the blade structure, under different loading
conditions. The most dangerous of these will be singled out as typical cases
calculated for structural strength.
Usually, in the development of new blades, when the time alloted for per-
forming and processing the calculations is liinited, it is desirable to reduce
the number of calculated cases to a minimum. Experience has shown that it siof-
fices to examine a single case of blade loading under ground operating conditions
of the helicopter and several cases in flight at different flight regimes .
The first case necessitates calculating a blade supported on the vertical
restrictor of the hub after full or partial stoppage of the effect of centrifugal
forces. This occurs when the rotor is not rotating or is in the initial stage
of overspeeding or else is stopped after the flight. In the absence of centri-
fugal forces, the gravitational forces or inertia forces arising ipon irrpact of
the blade against the coning stop set vp appreciable bending stresses. In this
case, conpressive stresses are especially dangerous for blade strength. Experi-
ments show that individual blade overloads, at which considerable conpressive
stresses are set xxp, may affect the fatigue strength of the structure and hence
its service life. Usually, static stresses due to bending of the blade under
the effect of its own weight are limited to values of Oq = 25 - 2S kg/mirf for .a
13
blade with a steel spar and of ag = 7*0 - 7»5 kg/mm^ for a iDlade -with a dural-uinin
spar.
From the conputational viewpoint, this case presents no difficulties;
therefore, we will not further discuss it here.
Other cases pertain to different helicopter flight regimes when constant
and varialDle stresses from "blade "bending are added to the permanent stresses
due to centrifugal forces. This combination of loads is highly detrimental to
the fatigue strength of the "blade structure.
3. FUght Regimes Detrimental to the Fatigue Stren^h /18
of the Structure
Ih-flight stress analyses have shown that helicopter "blades are subject to
appreciable alternate loads having a detrimental effect on the structural
strength in two different types of fUght regimes.
The first tjpe of fUght regime includes low-speed modes, when the flying
speed is 3 - 8^ of the blade tip speed (fj, = 0.03 - 0.08). In these regimes
there is a marked increase in the flexural vibration airpHtudes of the blades,
causing a corresponding increase in the variable stresses.
The helicopter uses the above range of flying speeds in acceleration, hori-
zontal fUghb at steady low speed, and in the braking regime. Usually the
greatest variable stresses arise in the braking regijue. Appreciable stresses
may arise also in a steep descent at low horizontal speed.
With respect to the conditions of loading of the structure, flights at low
speeds generally are short-term regimes, at least for helicopters used for
transport missions. However, because of the high stresses present, it is pre-
cisely these regimes that often determine the service life of the blade with
respect to fatigue.
The second type of regime detrimental to fatigue strength has to do with
high-speed modes. These conprise primarily flights at cruising and maxunian
speeds. A flight at cruising speed is usually the longest flight mode and thus
inposes considerable fatigue stresses on the structure.
A marked increase in variable stresses at low speeds can be attributed
primarily to the appreciable nonuniformity of the induced velocity field created,
during these regimes, in the flow through the rotor. Moreover, in absolute
magnitude, the induced velocities here reach maxim-um values in conparison with
all other flight modes. Therefore, their influence on the magnitude of stresses
increases greatly at low speeds. The variable induced velocity field leads to
variable aerodynamic blade loading. Under the effect of these loads the blade
executes flexural vibrations which set xxp considerable variable stresses.
At high flying speeds, variable aerodynamic loads are generated mainly as a
consequence of fluctiiations in the relative flow velocity and changes in angles
of attack of the blade sections with respect to the rotor azimuth. The variable
14
induced velocity field in these regimes has Uttle effect on the magnitudes of
the aerodynamic load.
In strength calculations it is sometimes necessary to allow for rotor over-
speeding which might occior in flight at a steep rise in centrifugal forces.
This will also cause an increase in the constant conponent of stresses in the
blade.
4. AssxHiption of a Uniform Induced Velocity Field
It is obvious from the alDOve that a calculation of variable aerodynamic
loads at low speeds is inpossible without consideration of the variable induced
velocity field.
On an increase in flying speed, the absolute magnitude of induced veloci-
ties decreases. The effect of their nonuniformity on the magnitudes of aerody-
namic loads also diminishes. Therefore, beginning with average flying speeds,
when |j. ^ 0.2, it can be approximately assumed in calculating variable blade
stresses that the induced velocity field is uniform, i.e., that the induced ve-
locities are constant over the rotor disk area. This assutrption leads to /19
significant simplifications of all computations and to a marked decrease in
calculation time. For this reason, it is widely used in practical calculations.
However, the accuracy of the results, with consideration of this assunp-
tion, often is unsatisfactory to the designer. Thus, it is often necessary to
abandon this assunption when calculating moderate and high-speed modes.
5. Assumptions in Calculating Aerodynamic Loads on the
Blade Profile
In all methods of calculation presented in this Chapter it is assumed that
aerodynamic forces acting on the blade profile can be determined by making use
of aerodynamic coefficients for steady flow past an infinitely long wing in a
plane-parallel stream. An unsteady state of the flow is taken into accoimt only
at values of the profile angles of attack at which downwash is introduced.
Consequently, to determine forces acting on a profile member, it is suffi-
cient to determine its angle of attack a and the relative velocity U of the flow
past it. Then, knowing a and M = — (where a^^ is the velocity of sound), we
^8
can determine from the profile polar the coefficients Cy and c^ and hence the
forces acting on the profile. If necessary, one can also determine the coeffi-
cient m^ .
If, in the flight mode under study, the profile angle of attack does not
exceed a « 9° and if the Mach number is not higher than M « 0-5, then we can dis-
regard its influence and assume that
15
Illllli
where c^ is the tangent of the angle of slope for the relation Cy = f(cy).
This assunption is used in calculating loads in flight modes sufficiently
far from separation in which, furthermore, we can disregard the coup res sibiHty
effect of the flow.
The possilDility of using various assimptions in the method of determining
aerodynamic forces is of great value in selecting the method of stress analysis
to be used in the case in question. As a consequence, it is suggested to use
different methods of calculation for different regimes • Below, we will differ-
entiate "between three types of regimes for each of which optimum results can "be
o"btained "by different methods of calculation* These are low- moderate-, and
high-speed modes.
In the low-speed mode, it is unavoidal)le to take account of the variable
induced velocity field but Hnear aerodynamics can "be used at average blade load-
ing. At moderate flying speeds, the variable induced velocity field need be
considered only in solving special problems raised by the necessity of differ-
entiating individual high harmonics of the aerodynamic loads. It is almost
always unnecessary at these speeds to consider nonlinear relations in determin-
ing the aerodynamic coefficients. Finally, in the high-speed mode which is
close to the separation limit, consideration of these nonlinearities becomes
mandatory, whereas the variability of the induced velocity field can be disre-
garded in most cases .
The above considerations result in individual methods of calculation tied /2Q
in with specific flight regimes.
6 . Relation of Deformations due to Bending in Two. Mutua lly
Perpendicular Directions and Corresponding Assumptions
for Calculation
Usually, a helicopter blade is designed such that the principal elastic
moments of inertia of its sections differ substantially in magnitude. Therefore,
the blade is considered as a bar extended by centrifugal forces, each portion of
which has different rigidities in two mutually perpendicular directions. To
characterize these directions, let us lay planes through the axis of the bar
along the direction of the principal axes of the section- which will be desig-
nated as planes of maximum and minimum rigidity (Rig.l.lj.
Frequently, to produce aerodynamic blade twist not only the frame forming
its contour is twisted but also its spar. In this case, the directions of the
principal elastic axes of the section vary over the length of the blade, chang-
ing it into a geometrically twisted bar. In other cases, aerodynamic twist is
obtained only by turning the frame of the blade relative to the spar.
In flight, external forces act on the blade profile in widely differing
directions. This changes the problem of blade bending into a highly cocplex
three-dimensional problem.
In addition, the degree of geometric twist of helicopter blades is only
16
Plane of
chord
Plane of minimum
rigidi ty
Plane of maximum
rigidity
flapping
Fig.l.l Position of the Spar at
Geometric Twist Obtained by Tiirning
the Frame Relative to the Spar
(cp^ = const) .
moderate (of the order of 6 - 12^) and
appreciably less than is feasible in
aircraft propellers or in cocpressor
and turbine blades. As shown by various
estimates, the effect of such twist on
the calculation results is only slight.
Therefore, in all the methods of calcu-
lation presented here we will disregard
the degree of twist of the elastic axes
of the blade spar and will assume that
the direction of the plane of maximum
and minimum blade rigidity is constant
over its length.
This assimption pennits projecting
all external forces onto these planes
and solving two elastically unrelated
two-dimensional problems of blade bend-
ing in two mutually perpendicular direc-
tions. After performing the stress analyses for various points of the blade
section, the results of both calculations can be simmaed.
The blade section profile permits increasing the size of the spar in the
chord plane and limits the chords in a perpendicular direction. Thus, the plane
of maximum rigidity is usually close to a plane passing through the blade chord.
This circumstance, as well as the fact that the magnitude of the aerodynamic
forces in the chord plane is usually smaller than in the plane perpendicular to
it, causes the magnitude of the bending stresses to be greater in the plane of
minimum rigidity and lower in the plane of maximum rigidity. A study of modern
blade designs, where the fatigue strength is approximately identical in omnidi-
rectional bending, indicates that bending in the plane of minimum rigidity is /2l
considerably more dangerous. In practice, all difficulties usually have to do
with the need of ensuring adequate bending strength in this plane. Therefore,
we will here discuss methods of calculating blade vibrations only in the plane
of minimum rigidity. For calculations in this plane, we can use the additional
assunptions that the plane of minimum rigidity coincides with the plane going
through the rotor axis. Below, we will designate this plane as the flapping
plane.
7. Consideration of Torsional Deformation of a Blade
in Calculations of F lexural Vibrations
Torsional deformations change the angles of attack of the blade sections
and hence the aerodynamic forces acting on them. Therefore, these deformations
should be taken into account in the calculation of aerodynamic loads and vibra-
tions of a blade. However, the consideration of torsional blade vibrations en-
tails considerable difficulties and greatly coirplicates the calculation.
At the same time, this does by no means always lead to substantially im-
proved results. Therefore, torsional deformation should be taken into account
only in cases of actual need, for exanple whenever the flextoral blade vibrations
17
Illllillli III
are anplxfied on approach to bending flutter; however, this in^^lies an iiiade-
quate margin of safety -with respect to flutter and must te considered inper-
missible .
To allow for torsional deformations, a system of differential equations of
bending-torsional "blade vibrations must be solved. Its solution is obtained by
calculation of flutter. Such a method of calculation, known as the general
method of calculation of blade flutter and bending stress, has been given in
the first volume of this book (Sect .7, Chapt.IV).
Here, we will describe only methods of calculating free torsional (Sect .5)
and bending-torsional vibrations (Sect*6).
8. Two Calculation Steps in Blade Design: Calculation of
Natural Vibration Frecfuency and Calculation of Stresses
If a newly designed helicopter blade does not differ excessively in geo-
metric and mass characteristics from an already manufactured and tested blade,
it can be asserted that in identical flight regimes the variable blade stresses
will be approximately the same as in the prototype blade. However, this rule is
violated when, as a consequence of some change in its parameters, the blade is
in resonance with some harmonic of the external forces.
Blade-design practice shows that siifficiently reliable blades can be de-
veloped only if none of its natural frequencies coincides with the frequencies
of the external forces and actually these are far apart. This pertains to blade
vibrations both in the plane of minimum rigidity and in that of maximum rigidity.
Naturally, it is obvious that not all harmonics of external forces, but only
those whose magnitude is siofficient to set up high stresses, are detrimental to
the strength of material. Usually, absence of resonance is mandatory for
harmonics not higher than the S'th relative to the rotor rpm. Higher harmonics
of external forces have little effect.
Thus, if a rough error in selecting the blade characteristics is iiipermis-
sible, variable stresses can be kept within permissible limits by preventing /22
the occurrence of resonance. In this case, there is no need to calculate the
variable stress airplitudes. Thus, the experimental designer can often Umit him-
self to the first stage of blade calculation: determination of its natural vi-
bration frequencies and plotting of the resonance diagram.
It follows from the above that the calculation of blade frequencies and
natural vibration modes is not only an airxiliary step in stress analysis but has
an independent value as a preliminary step in blade strength calculations.
9. Idealized Blade Models Used in Calculation
In performing the calculation, the blade must be represented as some ideal-
ized mechanical model for which all adopted initial assunptions would hold, so
that later - during the calculations - there would be no need to use approximate
mathematical operations.
18
With calculation on conputers, the problem should be prograinmed such that
its solution becomes possible with any prescribed accuracy attainable by the
conputer.
As shown by practical experience, calculation methods utilizing approximate
mathematical operations often lead to other misconcepts. In many cases, it is
impossible to conplete the calculation because of some inaccuracy in the conpu-
tations. For exanple, in calculating the natural vibration modes by the method
of successive approximations an entire series of integrals must be calculated.
This is often done by the trapezoidal method • At a liinited number of integra-
tion intervals, this method results in such a large error that, in calculating
the vibration modes of higher harmonics whose ordinates are calculated in the
form of small differences of large quantities, the method of successive approxi-
mations ceases to converge ♦
This fact necessitates special caution in using approximate methods of
calculation. Consequently, it is preferable to introduce a simplified idealized
blade model which could be calculated at maximxam permissible accuracy on the
conputer.
Three different types of mechanical models are known, which are frequently
used in calculations.
Beam model with continuously distributed parajneters . In this model, the
blade is represented as a beam with continuously distributed rigidities EI,
linear mass m, and parameters detennining the magnitude of the linear aerody-
namic load.
Such a model is highly convenient in deriving initial differential equa-
tions and in applying known approximate solution methods to them tut is unsioit-
able for performing numerical calculations. Below, we will frequently refer to
such a model in deriving working formulas so that, in the stage of niomerical
calculation, we can use formulas derived by analogy and pertaining to a model
with discrete parameters. In these formulas, all integrals of functions depend-
ing on the blade radius are replaced by the s"ums of discrete quantities pertain-
ing to a series of fixed blade radii.
Beam model with concentrated weights . In this model, the blade is repre-
sented as a system of coipled concentrated weights. The coipling between these
weights is acconplished by small weightless beams having a longitudinal constant
flexural rigidity equal to the rigidity of the corresponding blade elements.
In determining the aerodynamic forces, it is assumed that to each weight
is attached a separate small wing whose area is equal to the area of the /23
corresponding blade element. Usually, it is assimed that the area is
^i=^(^i-i.i + A-./^i)^,. (1.2)
where Ij-ii and Ij i+i = lengths of adjacent segments into which the blade is
divided in the calculation;
bi = blade chord in the section between these segments.
19
This model most accurately reflects the properties of a real blade. For
this reason, it will "be used in practical calculations in aHmost all cases.
However, we should mention that the beam model has these favorable proper-
ties only if the number of parts z is equal to 25 - 30 or more. As soon as this
number decreases, the type of deformations of the beam model begins differing
greatly from that of the deformations of the blade. This will be illustrated
in more detail in Section 10, Subsection 3* Furthermore, the use of the beam
model often leads to a rather conplicated system of equations and at times even
interferes with the calculation. In such cases, the sinpler hinge blade model
can be used.
Hinge model of blade . In this model, the blade is represented as a multi-
hinge chain consisting of absolutely rigid weightless links whose masses are
concentrated in the hinges. The flexural rigidity of the blade is simulated by
elastic members concentrated in the hinges. Under the action of external forces,
the axis of this chain takes the form of a broken line rather than of a smooth
line as in the beam^type model. This fact, just as the task of selecting the
rigidity of the elastic members, introduces a certain error when changing from
a blade to a mechanical model.
At the same time, the use of the hinge model leads to such great sinplifi-
cation of the working formulas that it often becomes possible to use inproved
methods of calculation which were not feasible when using the beam model. This
conpensates the faults inherent to this model.
It should be mentioned that, on a decrease in the number of segments into
which the blade is separated in the calculation, the properties of the models
begin to differ markedly from the properties of a real blade. However, for the
hinge model these errors do not increase as rapidly as for the beam model. As
a consequence, the hinge model may be more sioitable in rough methods of calcula-
tion, when the blade is divided into a small number of segments, say of the
order of 10-12.
10. Derivation of the Diff_erential Equation of Blade. Bending
in a Centrifugal Force Field at Vibrations in the
Flapping Plane
Let us represent the blade as a beam with continuously distributed para-
meters. For our study, let us isolate an element of the beam of length dr. The
forces acting on this element are plotted in Fig. 1.2.
Let us then construct the equation of equilibrixom of this element, limiting
the calculation to values of the first order of smallness. Then, the sum of the
projections of the forces onto the y-axis can be written as
Wdr^dQ^O^ (1.3)
and the sum of the moments of all forces relative to the point A
Qdr^dM—Ndy = 0, (1.4)
20
where
W = linear external load on the
blade;
Q = shearing force in the blade
section;
M = bending moment; /2^
N = centrifugal force in the
blade section.
From eq.(1.3) we obtain
(1.5)
Fig. 1.2 Diagram of Forces Acting
on a Blade Element.
Here and below the prime denotes
differentiation with respect to the
blade radius •
After differentiation of eq.(1.4),
we obtain
Q'—M''^[Ny'Y.
(1-6)
Setting M = Ely'' and substituting eq.(1.6) into eq.(l.5), we obtain the
known differential equation of bending deformations of a blade in a centrifugal
force field:
{EIt/r~[Ny'Y^W.
(1.7)
let us represent the external load W, consisting of aerodynamic and inertia
loads, in the form
W=T~my,
(1.8)
where
T = linear aerodynamic load;
m = linear mass of the blade.
Here, the two dots denote differentiation with respect to time.
After substituting eq.(l.B) into eq.(1.7), we obtain the differential
equation of blade vibrations
l/:'/t/'
Wy'V ~rmy-^-
(1.9)
In a vacuum, when the aerodynamic load T is equal to zero, eq.(l-9) will
describe free blade vibrations in a centrifxogal force field:
The solution of this equation offers certain difficulties. For this reason,
21
..!»
Section 2 will first give its solution for the case N = pertaining to a non-
rotating "blade.
11. Differential Equation of Blade Bending; in the
Rotor Plane of Rotation
On tending of the "blade in the plane of rotation, owing to concentricity
of the centrifugal force field, the tlade element will "be su"bject to an addi-
tional force which did not enter the equations in the flapping plane. With con-
sideration of this, eq.(1.8) should iDe rewritten in the form
W^Q-i-di^mx—mx, (1.11)
where
Q = aerodynamic force in the plane of rotation;
X == displacement of the "blade elements in the plane of rotation.
After substituting eq.(l.ll) into an equation analogous to eq.(1.7) "but /25
written for the plane of rotation, we o*btain the differential equation of blade
"bending in this plane
[EIXV-[^^'y-'^^^^-i'^^^=^Q^ (1.12)
This equation differs from eq.(1.9) only "by the additional term cu^mx.
Section 2. Free Vi"brations of the Blade of a Nonrotating Rotor
1. Method of Calculation for Solution of the Integral
Ecfuation of Blade Vibrations
Calculation of the natural vibration modes and frequencies of the blade of
a nonrotating rotor has been extensively described in the literature [see, for
exairple (Ref .1)]. In this Section, we will briefly repeat certain fundamental
premises and somewhat refine the formulas used for practical calculations.
Let us examine the differential equation of vibrations derived for the model
of a blade with continuously distributed parameters. If we set N = in
eq.(l.lO), it will take the form
[BIyT + m.y=0. (2.1)
Setting
and substituting into eq.(2.l), we obtain
[Ery"]"-p'm-^=0. (2.3)
In our further confutations, we will omit the vincviluin over y. Let us
22
integrate eq.(2.3) "with consideration of the boundary conditions of the blade
attachment. For sinplicity, let us take the case of a blade rigidly attached
at the root, with the following boundary conditions :
at r = 0; y = 0; y' = 0;
at r - R; M = 0; Q = 0.
By quadruple integration, eq.(2.3) is transformed into an integral equation
of the form
^-''{{wll'^y''-'- (2.4)
r r
Equation (2.4) is solved by the conventional method of successive approxi-
mations. Prescribing an arbitrary form of y, normalized in some manner, for
exatrple
(2.5)
let us substitute it into the right-'hand side of eq.(2.4).
After integration, we obtain a function
such that y = p^u.
2 ^
From this, using the condition (2.5), we obtain /26
(2.7)
where Ur is the value of u at r = R.
We then repeat the same operation, taking the new value
y^p^u. (2.8)
After carrying out the above operations several times, it will be found
that the vibration mode y and the frequency p converge to definite values which
constitute the solution of the integral equations (2.4)*
The method of successive approximations, applied in thj.s manner, yields a
determinable mode y converging to the mode of the lower harmonic of the natural
blade vibrations.
To determine the subsequent harmonics, it is necessary to satisfy the con-
dition of orthogonality of the natural vibration overtones. This condition will
be discussed in Subsection 3*
In practical application of the calculation method presented here, it is
inportant to select a sxofficiently exact method for calculating the integral
equation (2.6). If the blade parameters are given in the form of continuous
23
fxinctions, then the siaplest method of calculation of the integrals (2*6) is the
trapezoidal method generally employed in such cases. However, as already indi-
cated above, in calculating higher vibration overtones the uncertainty introduced
by this operation leads to such extensive errors that the method becomes useless
for practical purposes. This drawback is eliminated if, in calculating the
integrals (2#6), we use the method obtained from a study of the mechanical model
of a blade with discretely distributed parameters.
2. Calculation of the Natural Vibration Modes and
Frequencies of a Blade Model with Discretely
Distributed Parameters
For the calculation, we will use a beam-type model with concentrated loads
(see Sect.l, Subsect.9)» For this, let us divide the blade into % segments.
The length of the individual segments can be different. The weight of the blade
is concentrated along the edges of these segments in the fonn of individual dis-
crete loads with mass m^ . The f lexural rigidity of the blade is represented by
a stepped curve, so that it remains constant over the length of each segment
(Fig. 1.3).
Just as in Subsection 1, we will first examine the case of a blade fixed
at the root. The operation defined by eq.(2.6) can be carried out exactly here.
Actually, let us use an arbitrary form of displacement of the loads of the
model y^ . Here, the system of discrete values of y^ (i = 0, 1. 2, 3, ... z
being the serial number of the concentrated loads of the model) will be desig-
nated as the mode of displacement. As above [see eq.(2#5)], we set y^, = 1. If
the displacements yj are known, we can deterinine the inertia forces of the loads
on their vibrations with a frequency p = 1. These are determined by the expres-
sion
^i=^iyi- (2.9)
Knowing the inertia forces, we can determine all bending moments by a system
of simple recursive formulas of the form
Mi-=li, i+i [Fi+i~ai+iMi^i—bi+iMi+2l (2.10)
where Ij 1+1 is the length of the blade section between the i-th and i+l-th con^
cent rated mass.
The coefficients ai and bj are determined by the formulas /27
A calculation of the bending moments by eqs.(2.l0) should start from the
end of the blade, first putting i = z - 1 and then equating the bending moments
M^ and M^^i to zero.
21,
■ I mil II ■■iiniiiiwii III I Ml III II
After defining the bending moments, it is easy to determine the tilade de-
formations • As alDOve, the blade deformations during vibrations with a frequency
p = 1 vdll be denoted by the symbol u.
£1
. r
^U
1/ \i ij u \b
=©==^=iS=^
Fig.l#3 Calculation Model of Blade.
The magnitude of these deformations is determined by recursive formulas of
the type
(2.11)
where
Here,
^'..-1
(2.12)
(2.13)
Calculation of the deformation Uj should begin from the blade root, after
setting Uq = 0, in conformity with the boundary conditions adopted here. All
quantities with negative subscript should also be equated to zero.
Thus, carrying out the operations (2.10) and (2.11), applicable to a beam
model with a discrete distribution of parameters, leads to calculation of exact
values of Ui #
After determining p^ in the same manner as before [see eq.(2.7)]
and using the new values
p^^^
yi=p^uu
(2.34)
(2.15)
25
II II ill
we repeat all operations until the method of successive approximations con?- /28
verges • Usually, the calculation is considered conpleted as soon as the dif-
ference in the values of y^ , in two successive approximations, is less than the
prescribed accuracy Sy •
3. Condition of Orthogonality and Calculation of Successive
Natural Vi"bration Harmonics
The above method of successive approximations permits a determination of
the lower harmonic of natural vibrations • In detennining the higher harmonics^
it is necessary to satisfy the conditions of independence of the vibrations with
respect to different harmonics.
Let us imagine that free blade vibrations in vacuijm occur simultaneously
with respect to two modes y^^^ and yj"^ . The vibration energy for each of the
modes can be determined separately from the ajiplitude values of the kinetic
energy"" :
i
(2.16)
On the other hand, the total energy of the system vibrating simultaneously
with respect to two modes can be determined from the airplitude value of the
total kinetic energy:
K^^^m, [Piy^/^ + p^y\^^]\ (2.17)
I
The system has this kinetic energy at that instant of time when the blade,
during vibration, passes the neutral position simultaneously with respect to the
two modes y^^^ and y^^"^^ . Owing to the difference in the values of the natural
vibration frequency, such a position arises relatively seldom, but can be easily
created artificially by prescribing the appropriate vibration phases at the
initial instant.
If the amplitudes with respect to each of the conponent modes of vibration
do not change in time, then their energy, determined by eqs.(2.l6), also remains
constant «
The total vibration energy should always be equal to the sum of the ener-
gies of the conponent motions, i.e.,
K^^K^-^K^. (2.IB)
^^ For simplicity, here and below the constant 1/2 is omitted in the values of the
kinetic and potential energy of vibrations.
26
As follows from eq#(2.17), this is possible only if
(2.19)
This condition is known as the condition of orthogonality of the natioral
vibration harmonics. A more rigorous derivation of this condition will be given
in Section 2 of Chapter II #
In calculating any j-th harmonic, all previous harmonics to which the sub-
script m = 0, 1, 2, ••♦, j - 1 corresponds, should already have been calculated.
To satisfy the condition of orthogonality in determining the mode of the /29
J-th harmonic by the method of successive approximations, we will represent the
unknown mode y^ ^ ^ as
y\J)^p2
■ 2 c^^/i-)
(2.20)
where yj ^ are pre-viously detennined natural vi'bration modes.
The constants Cm are determined from the condition of orthogonality (2.19)
by the formulas
:S '"/"/I/i""
C„=
:s-/[i/i'"'j^
(2.21)
The value of the frequency of the j-th harmonic is calculated from
1
P)=-
m=0
(2.22)
Knowing p^, we can determine the vibration mode from eq.(2.20).
4. Characteristics of Calculation of Natural Vibration
Fr^cruenc ies and Modes of a Hinged Blade
All above-presented confutations pertaining to a rigidly fixed bIU.de can
easily be extended to a blade with hinge attachment at the root.
For this case, the integral equation (2.4) takes the following form:
y=.p^
|^JJ.,^r3+C,r
(2.23)
27
where the constant Cq is determined from the condition of equating to zero the
sum of the moments of all inertia forces relative to the hinge. For a model
with a discrete distribution of the parameters, this condition can "be written as
S'^'^/C— '■o)=0- (2.24)
i
It is olDvious that this condition satisfies the condition of orthogonality
to the vibration mode, which we will tentatively call the fundamental vibration
mode. If this mode is normalized in conformity with the condition (2.5), then
it can be written as
y(0)^r-ro_^ (2.25)
R — ro
Thus, in calculating a hinged blade it must be taken into account that the
mode of its fundamental is known beforehand and is prescribed by means of
eq.(2.25) and, in calculating all subsequent harmonics beginning with the first,
it is also necessaiy to satisfy the condition of orthogonality to the funda-
mental {2*2k) * Here, we can determine the functions u^ by the same formulas as
those given in Subsection 2.
5. Calculation of the Natural Vibration Modes and Frecfuencies /30
of a Blade as a Simply Supported Beam
It frequently is necessary to calculate the frequency of synchronous vi-
brations of the blade and helicopter fuselage. In this case the rotor hub it-
self, being the point of attachment of the blade, may be displaced together with
the helicopter fuselage. The calculations of such vibrations are very easy to
perform when using a blade model representing a siii5)ly supported beam. Then,
in determining the synchronous vibrations of rotor and fuselage, it suffices to
calculate the mass of the fuselage m^ reduced to the rotor (see RLg.1.3) and
then calculate the natural vibration frequencies of the blade.
Calculation of the blade as a siiiply supported beam can be performed by the
formulas given in Subsection 2, except that all natural vibration modes should
be additionally orthogonalized to the mode of the second fiondamental:
i/;o^ = l-corst, (2.26)
which is eqidvalent to satisfying the condition of equating to z ro the sum of
all inertia forces acting during the vibrations.
This method of calculation, with slight refinements, can be used also for
calculating the natural vibration modes and frequencies of the fuselage, which
will be taken up in Chapter II.
28
Section 3- Approximate Method of Determining the
Natural Blade Vibration Frequencies in
a CenbrifTigal Force FieH
1. Use of B,«G.GalerldLn's _Method for Determining the
Natural Blade Vibration Frecaiencies
The method of B.G.Galerkin is vadely used for solving various problems of
elastic blade vibrations.
The idea of Galerkin's method and its application to the solution of dif-
ferential equations is rather thoro-ughly covered in the literature [see;, for
exanple, the manual "Mashinostroyeniye" (Mechanical Engineering), Vol.1, Book 1,
Mashgiz, 19473 •
Here, we will not repeat conclusions that can be found in other sources
but will illustrate the use of this method on a number of siirple exanples.
In Subsection 10 of Section 1 in this Chapter, we derived a differential
equation of blade vibrations in a centrifugal force field. On substituting into
it the quantity y in the form of eq.(2.2)^ then this equation takes the following
form (we have omitted here the vinculum of y) :
[EIyT-[Ny^Y--p^my = 0. (3-1)
Let us assume that the natural blade vibration modes in a centrifugal force
field do not differ from the corresponding modes calculated for the case N = 0.
Then, taking into account that the vibration modes y^^ ^ are known, let us sub-
stitute some mode y^-' ^ into eq.(3.l) and^ after multiplying all terms of the
equation by this same mode y^ -* , integrate the obtained expressions over the
blade length.
The obtained equation, after certain transformations, can be represented
in the form
f£/[(i/TP^/- + jA^[(^^)T^/--/;^Jm(i/02^r=a . (3.2)
The integrals entering this equation /31
C^,=^]EI[{yr?dr, (3-3)
have a well-defined physical meaning, namely:
Cgi = elastic potential energy accumulated by the blade as soon as,
during flexural vibrations with respect to the mode of the j-th
29
harmonic, the t)lade shows extreme deflections from the eguillbriim
position--;
C|sj = potential energy accumulated "by the "blade while trending in a
centrifugal force field. Here, just as in eq.(3*3), different
harmordcs of the natural vilbrations can be studied.
The total potential energy accumulated "by the bilade while bending in a
centrifugal force field according to the formula of j^^^ can be written as
Cz^C^j + C;,, (3.5)
In flexural vibrations when the blade passes through the equilibrium posi-
tion, the rate of displacement of its points reach maximum values:
'y^J' = py'^^. (3.6)
In this case, the kinetic energy of the blade can be determined by the
formula
In free vibrations, the potential energy accumulated by the blade while
bending with respect to the mode y^^ is converted into kinetic energy when the
blade passes the equiUbri'um position. The equality of the anplitude values of
the potential and kinetic energy of the blade is expressed by eq.(3»2)»
From eq.(3»2), the frequency of the j-th harmonic of natural blade vibra-
tions in a centrifxogal force field can be obtained. This frequency is deter-
mined by the foi^ula
P'=Ph + ''j'''^ (3.8)
where
Pqj = natural vibration frequency of the blade without consideration of
centrifugal forces;
kj = a coefficient allowing for the effect of centrifugal forces.
Here,
Ph = '~ • (3.9)
j « [y^]-
'?dr
* This holds mth an accuracy to -within a constant equal to 1/2, which is
omitted in e<^.(3.3), (3.4)., and (3.7).
30
I IB ■
dr (3.10)
^m{y^\2c
In eq.(3*lO), N^jj^^^ is the centrifugal force in the "blade section at uu = 1. /32
Equation (3»9) for the natiiral vibration frequency vdthout consideration of
centrifugal forces can be obtained if the method of B.G.Galerkin is applied to
eq.(2»3) in the same manner.
The ejxpressions derived here for the natural blade vibration frequency in
a centrifugal force field are approximate. However, calculations show that, in
many cases, these e:xpressions give an accuracy conpletely satisfactory for
practical purposes. A more thorough evaluation of the accuracy of the results
of these calculations will be given in Section 4*
2. Resonan c^_Dia^rajii of Blade Vibrations
As mentioned above, in blade designing calculations are required to pre-
clude possible resonances of natural blade vibration frequencies with the
harmonics of external forces, which might set vsp appreciable variable stresses.
As stated before, the harmonic conponents of aerodynamic forces acting on a
blade in flight are of substantial magnitude, ip to harmonics not exceeding the
S'th. Higher harmonics of aerodynamic forces are so small in magnitude that they
can be disregarded.
The frequencies of forced vibrations, which are a soiu?ce of concern in
blade calculations, can be determined by means of the formula
v=/^(o, (3.11)
where n = 1, 2, 3, ..., 8.
Equation (3»B) permits constructing the dependence of natural vibration
frequencies of various hai^onics on the angular velocity of rotation of the
rotor. Equations (3»8) and (3»ll), plotted jointly on one graph, are usually
called the blade resonance diagram. Figures 1.4 and 1.5 give resonance diagrams
constructed for blades with different parameters encountered in practice. These
diagrams are plotted in relative values. Both the natural vibration frequency p
and the rotor rpm refer to a certain operating value of the rpm^n^p .
The resonance diagram permits tracing, in graphic form, the direction
toward which the blade parameters should be changed so as to eliminate resonance
in the entire range of operating rotor rpm.
31
3. Selection of .Blade Paraineters to EliTninate Eesonance
during Vilpration in the Flappirig: Plan e
A scrutiny of the resonance diagrams, constructed for the diverse "blades,
shows that they do not differ greatly* The existing difference can mostly be
attributed to the difference in the flexural blade rigidity. Less often and to
a lesser degree, the cause is a deviation in the blade mass characteristics.
This can be explained in a sinple manner. Actxially, the designer must be guided
by a large number of various requirements, which limit the possibilities of vary-
ing the blade parameters and ultimately J^^d to the creation of blades with
closely adjacent characteristics.
The following conditions place the main restriction on extensive variations
in blade parameters:
1. The height of the spar is limited by the blade profile and cannot be
increased much, since an increase in relative profile thickness will automatical-
ly deteriorate the L/D ratio of the rotor. This places an \pper limit on the
magnitude of flexural rigidity of the blade. 121
2. The bending deflection of the blade under its own weight should not be
excessive, since it will lead
"v to difficulties in laying out
the helicopter. Bending
stresses in the spar, set ip
by dead weight, should not
exceed known magnitudes which
are selected from strength
conditions with consideration
of possible dynamic over load-
ings. These considerations
limit the possibilities of
reduction in blade rigidity.
3. The weight of the
blade is confined within even
closer Umits . The endeavor
to increase the weight factor
of a helicopter forces the
designer to reduce the blade
weight to a minimxmi. However,
this leads to an increase in
variable stresses due to bend-
ing, acting in the blade during
flight and hence leading to a
decrease in its service life.
Therefore, the blade weight
usually is decreased until the
spar starts being subject to
increasing variable stresses.
As a result, blade weight is
strictly dependent on rotor
10 n/npp
Fig. 1.4 Resonance Diagrams of Various Types
of Blades in the Flapping Plane.
32
size and on the strengi^h characteristics of the material from which the rotor
spar is fabricated.
As a consequence, the resonance diagrams of different blades vary in
practice within Umits that are "bounded on one hand "by the feasibility of a
highly rigid blade and on the other by the feasibility of an adequate service
Hf e of low-rigidity blades .
For a given total struct -ural weight, a blade of maximal rigidity is ob- 734
tained if the spar material is arranged along the contoior of the profile, i.e.,
if the spar is inscribed in the blade profile. In this case a large percentage
of the blade weight can be put into its power member, the spar. Such blades
usually are most advantageous from the aspect of magnitude of effective stresses,
but they are difficult to manufacture. Blades with a free form of the spar
cross section (for exanple, of tubular shape) which are not inscribed in the
blade profile are sinpler to manufacture. However, such blades have little
resistance to bending and provide the least favorable resonance diagram during
vibrations in the flapping plane.
The following blade types can be distinguished with respect to dynamic
characteristics in the flapping plane:
Blades of low rigidity in the flapping plane . Such blades are usually em-
ployed in a structure made of tubular steel spars, with a frame not subject to
bending. In Fig. 1.4 the broken line shows the resonance diagrams for a blade
whose rigidity in the flapping plane is at the lower Umit of rigidity encount-
ered in practice. With such parameters, the blade enters into resonance of the
second tone with the foiorth harmonic and of the third tone with the sixth har-
monic of the exciting forces, which is the reason for the creation of appreciable
stresses of the same frequencies (see also Fig. 1.66). These resonances are
especially manifest at low speeds where the stresses of a blade of this type are
even higher than at maximum speed (Fig.l.64)» Therefore their service life, as
a rule, is limited by the length of their stay in low-speed modes.
Blades of low rigidity are usually unfavorable with respect to strength and
service life but are often used since they are the easiest to manufacture.
Blades of moderate rigidity in the flapping plane . With an increase in
rigidity, the natural vibration frequencies of the blade move away from these
resonances. This permits the designing of rather successful blades- In Fig. 1.4
the resonance diagram of this blade is shown as a solid line. As follows from
the diagram, the second tone of vibrations of such a blade has still not ap-
proached the fifth harmonic, while the third tone was somewhere between the
seventh and eighth harmonics. Designwise, these are usually blades with a con-
to-ur (or close to this shape) spar inscribed in the profile. The spar can be
either steel or duralumin.
It is iiipossible to obtain a further increase in rigidity without increasing
the blade weight. Moreover, even a slight increase in rigidity may lead to
resonance of the second tone with the fifth harmonic of the external forces.
Therefore, only heavy blades of greatly increased rigidity can be the next pos-
sible type in the sequence of increasing rigidity.
33
Heavy^blades of hig:h rigidity in thg flapping: plane « In increasing the
weight of a given t>lade, putting this weight into the structure of the spar, the
rigidity can "be increased so much that the frequency of the second tone will be
ahove the fifth harmonic. In this case, the resoriance diagram shown in Fig. 1.4
"by the dot-dash line is possil^le* lower variable stresses will act in the spar
of a blade with this resonance diagram, but the blades will be somewhat heavier
in conparison with blades of moderate rigidity. However, for small helicopters
in which the relative rotor weight is low, such an increase in blade weight is
feasible .
It should be noted that, in evaluating the dynamic characteristics of vari-
ous blades in the flapping plane, the position of the first tone of blade vibra-
tion has been conpletely disregarded. Usually the first tone lies between .the
second and third harmonics and its location can be changed substantially only /35
in structures differing by some special features, for exanple, jet rotors with
engines on the blade tip or rotors with nonhinged blades. The negligible dis-
placement in natural frequency of the first tone observed for ordinary rotors
generally does not greatly affect the magnitudes of the effective variable
stresses.
4 . Selection of Blade Parajiiete rs t o_E3imimt e Re s onanc e s
in the Plane, of Rotation
In designing a blade, absence of resonance must be ensured also in the
plane of maximum blade rigidity, which can be approximately considered to coin-
cide with the plane of rotation of the rotor. The plane of maxLmimi blade rigidi-
ty losually coincides with the plane of the chord. Therefore, the rigidity char-
acteristics of a blade in this plane may vary in wider limits than in the flap-
ping plane. Beginning with a circular tube, the cross section of the spar can
increase to a size occupying practically the entire profile from the leading to
the trailing edge. However, there are certain limitations also in this plane.
Thus, an increase in the width of the spar with respect to the chord is certain
to lead to a shift in blade centering toward the trailing edge, which is usually
iri^^ermissible from the viewpoint of requirements for the prevention of flutter.
Furthermore, an increase in width of the spar may be acconpanied by an increase
in variable stresses. A decrease in rigidity of the spar by reduction of its
width automatically leads to a decrease in torsional rigidity of the blade.
This constitutes one of the factors preventing the creation of blades of low
rigidity in the plane of rotation.
In evaluating the resonance characteristics in the plane of rotation it is
mainly necessary to investigate the first and sometimes also the second har-
monic of blade vibration. The excitation of vibrations by higher harmonics is
not as likely.
Blades can be subdivided into the following types, based on their dynamic
characteristics in the plane of maxim-urn rigidity:
Blades of minimum rigidity in the plane of rotation . This type of blade
usually includes those with a tubular spar and a frame not subject to bending.
The natiiral vibration frequencies of this type of blade in the plane of rotation
34
are approximately the same as in the thrust plane or even somewhat lower, due to
the fact that the value of the coefficient kj [see eq.(3^S)] in the plane in
question is somewhat lower (this will tie taken vp in Sect .4, Sut)sect.4)* The
first harmonic vibrations in this case is generally somewhat higher than the
second harmonic of external forces so that no serious trout>le is created by this
resonance- The situation becomes worse for the second harmonic. This might
enter into resonance with the fourth harmonic of external forces. Generally,
this leads to a substantial increase in stresses of this particular frequency
in the plane of rotation. In Pig. 1.5 the dashed line represents the resonance
diagram for a blade whose rigidity in the plane of rotation Hes at the lower
Umit of rigidity encountered in practice. This blade is close to resonance of
the second harmonic with the fourth harmonic of external forces.
EHjides of low rigidity i n the plane of rotation . If the rigidity of a
blade in the plane of rotation is somewhat increased, so that its first tone re-
mains between the second and third harmonics and the second tone gets out of
resonance with the fourth harmonic, then the blade will be adequate with respect
to stresses in the plane of rotation. It should be noted that, with an increase
in rigidity, resonance of the second tone with the fifth harmonic to the rotor /36
ipm must be prevented.
Practice has shown that, at
this resonance, the stresses
in the plane of rotation in-
crease rather strongly,
which might even affect their
service life. The resonance
diagram of blades of low
rigidity in the plane of ro-
tation, for which the second
tone is between the fifth
and sixth harmonics, is
shown in Fig.1.5 by solid
lines.
Blades of low rigidity
in the plane of rotation are
widely used in practice, and
as a rule, cause no troubles
associated with vibrations
in this plane. However,
their rigidity characteristics
in the plane of flapping are
often close to those for
blades of low rigidity in the
flapping plane, which are
distinguished by high
stresses at low flying
speeds. On increasing the
blade rigidity in the flap-
ping plane, the rigidity in
the plane of rotation often
is simultaneously increased.
Fig.1.5
Resonance Diagrams of Various Types of
Blades in the Plane of Rotation.
35
This makes it necessary to use blades of even greater rigidity in the plane of
rotation*
Blades of moderate and high rigidity in the plane of rotation * Blades of
moderate rigidity in the plane of rotation usually include those whose funda-
mental Hes loetween the third and fourth harmonics of external forces, while the
second tone is located in a frequency range with such weak excitations that it
can "be disregarded* In Fig*1.5, the frequency of the fundamental of these blades
is shown by a double Une.
Blades of high rigidity in the plane of rotation include those whose /37
frequency in the fundamental Hes above the fourth harmonic of external forces
(dot-dash Hne in Fig.1.5).
Blades of moderate and high rigidity in the plane of rotation can be fabri-
cated with moderate stresses. However, in the practical use of such blades dif-
ficulties often arise, associated with a decrease in blade frequency as a conse-
quence of elasticity of the rotor attachment point to the fuselage. This must
definitely be taken into account in designing blades of this type.
Section 4» Calculation of Natural Blade Vibration Modes
and Frequencies in a Centrifugal Force^ Keld
1. Purpose and Problems of Calculation
As mentioned in Section 1, Subsection 8, the natural vibration modes and
frequencies of the blade must be determined in solving two types of technical
problems that inpose different demands on the method of calculation.
The first type includes problems in which the calculation of modes and fre-
quencies is carried out to select blade parameters that will prevent the appear-
ance of resonance. In this case, the calculation is con^leted by construction
of the resonance diagrams, and the natural vibration modes play only the role of
intermediate results and are not used later. Therefore, in current calculations
of this type, the natural vibration mode of a given blade in a centrifugal force
field is assumed to coincide with the mode of a nonrotating blade. The effect
of centrifugal forces is taken into acco-unt only in the values of frequencies
conputed from energy relationships determined by eq.(3.8). Such a rather sitrple
method of calculation is fully adequate for the purposes involved.
The second type includes problems in which the natural vibration modes and
frequencies are used for calculating forced vibrations, with a determination of
variable stresses set up in the blade structure. To obtain sufficiently accurate
results here, it is inportant to allow for characteristics that introduce tensile
centrifi:igal forces into the vibration mode.
It will be shown in this Section that centrifiogal forces substantially
change the natural vibration mode of the blade. The effect of centrifugal
forces is especially manifest in the form of curvature distribution of an elastic
line over the blade length and, to a lesser extent, in the mode of displacement
of its elements. A change in the form of cTorvature distribution naturally leads
36
to a redistrilDution of bending stresses over the blade. The effect of centri-
fugal forces on the distribution of stresses over the blade length is felt most
at sites of a marked drop in flexural rigidity and at sites of concentrated
loadings .
It should be noted that, in determining the natural vibration modes with
consideration of centrifugal forces, certain difficulties are encountered that
must be examined in greater detail.
2* Limits of A pplicabilLty of Calculation Methods Reducing
to a Solution of the Integral Equation of Blade Vibrations
To calculate the free vibrations of a blade in a centrifugal force field,
it is convenient to use the same method as for the blades of a nonrotating rotor.
However, the method of successive approximations (see Sect .2), which involves
solving the integral equation (2»1), cannot be applied in all cases to the soli>-
tion of eq.(3.l) describing natural blade vibrations in a centrifugal force /38
field.
It was shown in Section 2, Subsection 1 that, with a fourfold integration
of eq.(2.l), the problem reduces to solving the integral equation (2.4)* This
equation can be written in a somewhat different form
R R
where Mij^^j.^ ^ / J mydr^ is the bending moment due to inertia forces arising -upon
r r
blade vibrations with a frequency p = 1.
In the same manner, on integrating eq.(3»l) the problem reduces to solving
an equation of the following form:
5 f f {^ Inert — y^cf ) ^^2
^=P\\ -ET^ • (4.3)
where M^.^^ is the bending moment due to centrifugal forces at an angular velocity
of rotation of the rotor uj = 1:
M
cf = ^myrcir~yjmrdr; (4.4)
r r
j^ (4.5)
The method of successive approximations applied to eq.(4«l) yields satis-
factory convergence in all cases of rotor calculation but, applied to eq.(4»3)*
it will converge only in a certain range of values of the parameter y .
37
|||l|| l|l||||ll|llll|ll||IHB ^WII^MMHI nil I iiiiiw ■^■^■^« ■ -I -'■' '
Illlllllli
Figiire 1.6 gives the resonance diagram for a conventional helicopter iDlade '^
with hinged attachment to the hub. In this graph, the rotor xpm is laid off on
the abscissa and the natural vibration frequencies on the ordinate.
The values for the natural frequencies, obtained by solving eq.(4»3) vdth
the method of successive approxxmations, are shown in Fig. 1.6 by dots* Opposite
each dot, we entered the corresponding value of the parameter y and the number
of approximations s necessary for achieving the required accuracy of 0#00l. The
graph indicates that, at certain values of y> the value of s begins to increase
rapidly and the method of successive approximations ceases to converge.
7th karmonic 6th harmonic
5th harmonic
Uth harmonic
3rd harmonic
2nd harmonic
1st harmonic
ZOO 240 n ppm
flange of operating
rpm
Fig. 1.6 Resonance Diagram of Helicopter Blade in the Thrust
Plane, Constructed by the Method of Successive %)proximations.
If follows from Fig. 1.6 that, in the operating ipm range for helicopter
blades, this method permits a determination of natural frequencies of the third
and higher harmonics but only if all harmonics of the vibrations are determined
for a constant value of the parameter y, which only approximately corresponds to
conditions of the formulated physical problem. If, in the process of successive
approximations, the parameter y is^ refined for a given value of angular velocity
u), then the method will converge only in an rpm range appreciably smaller than
the operating rpm.
38
This requires the lose of other methods that afford a more relia'ble result
in the entire rpm range of the rotor.
3* Possible Methods of Calculating F ^ ee Blade Viljrations
in a Centrifugal Force Field
/39
Various methods can be used for calculating the natural vibration frequen-
cies and modes in a centrifugal force field. Of Soviet work, published on this
subject matter, we should mention three papers (Ref.4, 8, lO). Papers were also
published in other countries (Ref .33, 34) • In these, a rather cumbersome method
is presented which, moreover, does not yield a high accuracy of the final
results despite the fact that the calculation should be carried out to not less
than the lO^h to l2"t'h significant figure.
Here, we will present a method which, in our opinion, is the most convenient
for calculating the natural vibration frequencies of the blade in a centrifugal
force field. The process is based' on the three-moment method used by T.Morris
and W.Tye (Ref -32) in calculating bending stresses in a blade extended by cen-
trifiogal forces. The Morris and Tye method is also presented elsewhere
(Ref.l2).
The three-moment method, applied to calculation of a blade extended by
centrifugal forces, has a number of significant advantages, the main one being
that it does not require a high accuracy in the calculation process. The calcu-
lations can even be carried out with an ordinary slide rule.
The three-moment method has long been in use for calculating natural /40
frequencies, and has been programmed on the electronic conputers "St re la" and
M-20. Calculation of the first eight harmonics of natioral vibrations takes only
about 3 minutes on the "Strela"
conputer. A large niomber of the
most diverse calculations have
been performed. The results indi-
cate the extreme convenience and
great reliability of this method.
It should be noted that, when
using a conputer program for such
a calculation, there is no need for
any sinplified methods of calcula-
tion, for exanple, those mentioned
in Section 3*
Fig. 1.7 Polygon of Forces Acting on Two
Adjacent Blade Elements.
4* Three-Moment Method for Calcu-
lating Natural Blade Vibration
Modes and Frequencies in a
Centrifugal Force Field
To derive the conputational
formulas, we used the blade beam model with concentrated loads, discussed
39
already in Section 2, Subsection 2» As "before, we present the flexural rigidity
of the blade as a stepped curve, so that it remains constant over the length of
each segment (see Fig*l»3)» We will asstime the centrifugal force to be applied
only to the loads # Therefore, this value will be constant over the length of
each segment. We will also asstime that the centrifugal force is absorbed by a
special attachment of zero weight, free to move vertically.
It is obvious that such an idealized calculation scheme will be reliable
if the number of segments z is taken as sufficiently large. Usually the blade
is divided into no less than 25 - 30 segments (elements).
The method proposed later consists in determining the natural oscillation
modes and frequencies of such an idealized scheme, without additional assijnp-
tions.
Let us examine, two adjacent blade segments, deflected under the effect of
inertia forces from the plane of rotation of the rotor (ELg.l.y). As usual, we
will examine only small deflections.
The equation of equilibrium of each of the segments under the effect of
forces external to the given segment can be written in the form of zero-equality
of the sum of the moments of all these forces relative to some point . In this
case, we must include in the sum of the moments of these forces the shearing
force Q and the bending moment M acting in the cross section.
Then, the sum of the moments of forces acting on the blade segment 0-1 Al
relative to the point can be written as
The Stan of the moments of forces acting on the segments 1-2 relative to
the point 1 reads
M^—Mi—Ni2{y2—yi) -hQi2/i2=0. (4,7)
Here,
z
Qoi^—^^tyi;
1
z
Qu^-^^iyr
2
After dividing eqs.(4»6) and (4«7), respectively, by loi^oi ^^ Us^is stnd
adding them, we obtain the following equation of equilibrium:
^yo+^iyi-\rl^iy2^rn,M, + n,M, + m,M,+^^^. (4.8)
^12 ^^01
The notations introduced here, as well as in eqs.(4*l2), (4*13)* (4»1^),
and (^•15)> are given below [see eqp.(4«18) - (4»25)]-
40
In the same manner as eq.(4»S), we can write the equations of equilibrium
for all other "blade segments-
Examining, as iisual, only small displacements of the "blade elements, we
first determine the deformations of the segment 1 - 2» The equation of deforma-
tions of the element 1-2 can "be written, as conventional [see eq-(3»l)]*
The inertia term is absent here, since inertia forces are applied only at
the "boundaries of the segment- Taking into consideration that EI =, const and
N = const over the length of the segment and also that Ely^' = M, we o"btain
^-'^?^==°- (4.9)
where
The solution of eq.(4*9) can be written in terms of hyperbolic functions,
in the following manner:
The coefficients A and B are found from the following boundary conditions:
f or X = Mx = Ml ;
for X = lisMx = Mg-
From this, it follows that
sinh ai tcffihai
where of^ = m,iIis •
Substituting these values into eq.(4-10), we obtain A2
M,
-^^-^''-[s^"ta-;^>"^^^^+^^^^ (4.11)
Twice integrating eq.(4«ll), assuming y' = Pi , y = yi at x = and y' = Pa,
7 = J'z at X = Ija, we ottain either
or (4.12)
h {y-i - i/i) = - e^Mi - diM^ + p2. J
The equation of deformations for the segment 0-1 can be written ty analogy
a
with the second equation of the system (4.12):
(4.13)
After changing all signs in eq.(4-l3) to the opposite and adding to the
first equation of the system (4*12), we olDtain
boyo+cLiyi'{-biy2-=^doMQ+CiMi-i-diM2.
(4-1^)
Su*bstituting the left-hand side of eq#(4.1^) ^ov the Trending moments into
the equation of equilibriim of the elements [eq.(4»S)], we obtain the following
equation:
N12 A^Ol
(4-15)
Repeating the calculations for other segments of the iDlade, we o"btain a
system of differential equations with respect to the unknown functions of time
yj and M^, which is written out "below.
This system, expressed in the form of tables, consists of two systems of
equations (4*16) and (4*17)^ each' of which conprises z + 1 equations.
Any of the equations occupying one row in Table 1.1 represents a polynomial
whose coefficients are entered in the squares. All terms of the polynomial re-
present the products of some coefficient determined "by eqs.(4.18), (4.21),
(4»23) and (4*24) - (4.27) as well as the unknown functions Mj and y^ or the
second derivative of y^ with respect to time.
Only the coefficients of these functions are entered in the squares of
Table 1.1 while the fimctions themselves, simultaneously entering several equa-
tions, are given in a separate row above the tables.
The described system of equations also includes equations of the type of
eq.(4.l2), pertaining only to the root and tip segments of the blade and con^
taining the boundary values Pq and p^ • These equations are needed for calculat-
ing the boundary value problems.
The obtained system of equations has the following form:
Table Z. 1
m
fio
f^o
M,
Ml
• «•
• ••
^i;
M,
A
1
9o
K
*,
^r
^
r
K
S>
'^z
• ••
• ••
• •«
^^-3
^^•^
*.-.
*^-^
7^.f
^.,
"2;
3^
1
A.
y'l
y^
y.
«••
■>i-;
yz
*01
*«
*<,,
•• •
h,i-t
«*,
s,
t,.
t,,
•••
^t.7-r
tu
^i
<«
« ••
h,,.,
^22
...
»»-7
^2J^I
*,....
*»-,
Kuz
*«
(4.16)
42
mil ■■ iHaiiiii
1 1 in Hill mil I III iiHiiii iii^Hi^ iiHiiii I
A
M,
M,
Mi
• • •
• • •
Mz;
M,
fiz
■
1
'd ^,
a.0
Hi
t,
i,
<^i
^z
• •«
• ••
• ••
<i.-7
"^-2
'^z-Z
<i..z
^1-1
<iz.,
^z;
<=z
-1
yo
y,
yz
• • •
^z'-Z
yz-i
yz
"■0
*.
h
a,
b,
*r
o-z
h
•
• ••
• ••
•••
h-s
"■z-z
"z.Z
t>2-Z
a^.,
"z-,
h-i
«J
(4.17)
The following notations were adopted in constructing the above equations;
1
^01
; ^0=— ^o;
1
/nn=-
nt/ =
1
^1= —^i'l — ^il
a,=0.
V Sinn Oq /
' * ^ sinh a/ / *
flf, = 0.
«,=0.
^0 — ^0»
(4.18)
(4.19)
(4.20)
(4.21)
/Ml
i.h.22)
(4.23)
43
Illllllllli
h,=di—mi;
■n,\
gi=cr
gz=Cz-n^-
{h.2k)
(4.25)
In eqs.(4»26) and {k-*Zl)t given telow, nij is the mass of the i-th load
(4.26)
So=0;
Nz-
z—\,z
••0/
^m,
(4.27)
( ^01 )'
Here, the subscript k denotes the number of the row in Table 1.1.
To solve the system of equations in Table 1.1, it is convenient to use the
method of successive approximations. Mth respect to this system of equations,
this involves the following: The functions of time y^ (t)^ M^Ct), and 3i(t)
entering into the systems (4-16) and (4-17) are represented in the following
form:
y i i^) ==^1 sin pi;
where the letters y^ , M^, and p^ now denote only ajipHtude values of these
functions •
Then, bearing in mind that y^ (t) = -p^yi sin pt and canceling for sin pt,
we obtain a system of algebraic equations analogous to the system (4*16) and
(4*17) • Only the values of p^ will appear on the right-hand sides of the system
of equations analogous to eq.(4»l6).
Let us begin the method of successive approximations after assigning some 745
function y^ as the zeroth approximation. The second subscript here denotes the
number of t?ie approximation. The function y^ taken as the zeroth approximation
should somehow be nonned, for exanple
44
y,^L (4-28)
If the function y^ is known, then the inertia forces entering the right-
hand side of eqs.(4»l6) can be deteirmined with an accuracy to within a constant
factor p^ •
For the time "being, we will assume p^ = 1. Then, eqs«(4*l6) will yield the
values of the tiending moments Mj and the angle of rotation of the blade at the
root Pq • Next, from the known values of Mj and Pq we can determine, over
eqs«(4*17), the displacements of the blade axis during deformation which, for
the case p? = 1, we will denote by Ui such that
yi^P^Ui. (4.29)
After determining the displacements u^, we can define the natural vibration
frequency p. Its value is obtained on the basis of eqs#(4*28) and (4*29) in the
following manner:
p'-^^-Z- (4.30)
tlz
Then, in conformity with eq.(4-29) we determine the refined (after the
first approximation) function
yu^-p^,^ (4.31)
The entire process is repeatea Tintil the required accuracy is achieved.
This method of successive approximations results in the determined mode y^
being reduced to the mode of the lower harmonic of the natioral blade vibrations.
In determining successive harmonics, the condition of orthogonality must be
satisfied. The operations required when obeying the condition of orthogonality
are the same as for a blade of a nonrotating rotor (se.e Subsect.3 of Sect. 2).
The above equations are equally suitable for calculating the natural vibra-
tion frequencies in the flapping plane and in the plane of rotation of the rotor.
When calculating in the plane of rotation, the above values of the frequencies
should be corrected by the formula
P^i.rot -4i.fi -<^^ (4.32)
where o) is the angular velocity of rotation of the rotor.
The method of calculating the natural vibration modes remains the same, ir-
respective of the plane in which the calculation is performed.
Let us make a more detailed study of certain operations in performing one
approximation*
45
■I III III
5. Determination of Bending Moments on the Basis of
Known Forces
Let us begin with a determination of the bending moments on the basis of
known inertia forces entering the right-hand side of eq.(4.l6), which we deter-
mine in each approximation, assigning at first the value p^ = 1.
After prescribing some vibration mode y^ , the coefficients of the right-
hand side of eqs.(4»l6) can be deterinined which will be denoted here by Fj^ .
The coefficients Fj^ can be determined from the formulas
or, still better, from
_ Qk-^.i
Qk.
ft+i
Nu^:
1. ft
N.
ft.A+l
(4.34)
where Q^_^^^ =E miyi-
Then, the system of equations (4-16) can be rewritten in the following
form (Table I.2):
Table 1.2
Ho ^0
M,
Mz
• • •
• • •
Mz.f
Mi
Hz
, ___ , ., „ _
J ,
1
^0
ko
— -1
flf'
■\ Hi
h
hi
• • •
• k •
• • •
hz.j
S^-^
K-z
,
1z-z
9z-i
k-r
—
^-7
\ Sz
~l
m^
P'F,
P%
P%-:
feli
piFt
>
(4.35)
To solve this system, we must know two additional equations which take the
boundary condition into consideration. These equations can be the following:
at rigid attachment of the blade root
Po=0;
at rigid attachment of the blade tip
P.=0.
At hinged attachment of the blade tips or with conopletely free tips, we-
have Mo = and M^ = 0.
46
Below, we -will discuss only the two most common cases where the blade tip
is free (M^ = O) while the root either has a hinged sipport (Mq = O) or a rigid
attachment (po = O).
Let us examine the first case in which the blade is hinged, i.e., Mq = 0.
Here, to determine the bending moments we use only the equations encased by a
soHd line in the system (4*35); from the first equation we can then determine
the value of the angle of blade rotation in the hinge 0^ • I^om the last equa-
tion of the system (4 •35) we could also determine the value of p^ • However, we
do not need this value for further solution. The equation itself is used only
if g^ = 0, a case rarely encountered in practice.
In solving the system (4*35)^ it may easily happen that the wrong path is
selected, leading to the appearance, during solution, of small differences of
large quantities, which might conpletely ruin the result even when using a com-
puter providing an accuracy to 10 decimal places.
We propose here a repeatedly verified procedure, which permits performing 742
the calculation on an ordinary sUde rule.
We divide the first equation of the system (4*35)^ written for a hinged
blade, by gi and the second equation by hi:
'"■+7r*'=z^ (4.36)
M,+fM,+!^M,-^. (4.37)
Al hi hi
Subtracting eq.(4»36) from eq.(4«37) and introducing the following notations:
sl=
-.M2.-
*
^1
//•=
A,'
Fl =
^1
>
we obtain an equation analogous to eq.(4»36):
In combination with the next equation of the system (4«35), this equation
forms a system of two equations analogous to eqs.(4*36) and (4»37)* Repeating
the described operations a certain number of times, we ultimately obtain one
equation of the following form:
^'-^=lE; ' (4.39)
47
Ilillll
After determining the moment M^-i, we determine the moment M^-g, and so on
vp to the moment Mi« In other words, the moment M^ is determined each time
when the moment M^^^ is already determined. The formula for determining the
moment Mj can tie written on the "basis of eqs»(4»36) and (4»38) in the following
manner:
After obtaining the "bending moments, the angle of rotation of the blade in
the root hinge g^ is determined by means of the formula
^o^Fo—hoMu (4.41)
The second step in the method of successive approximations involves deter-
mining the blade deformations from known values of the bending moments Mj and
the angle of blade rotation in the hinge g^ •
6. Determination of Displacements from Known Bending Moments
Displacements of the blade in deformation which - in conformity with the
above - are denoted by u^ can be determined from the system (4«17)» However,
it can be demonstrated that the equations of the system (4»17) are inadequate
for determining all values of u^ .
Actually, for determining the position of the curve at a known curvature /48
distribution over the length, when given the values of Mj, and at a known value
of the angle of rotation at one point p^, one more condition inposed on the
values of displacements is necessary. In this case, the last equation of the
system (4»17)* which incorporates the value of the angle of rotation at another
point B2 > is actually identical with the first equation so that it can be
written out exclusively by analogy with the system (4*16) •
Thus, the auxiliary condition either will be the condition
"''=0' (4.42)
if there is a sipport at the blade root, or else the condition
z
2'«/«/=0' (4.43)
if the blade is regarded as free on two sides of the beam. The condition (4.43)
coincides with the expression obtained from the condition of orthogonality with
the fundajnental of the vibrations
i/(o) = l=const.
48
Calculating the coefficients that conprise the already determined values
of Ml aixl 0Q and leaving only the first of the two identical equations, we ob-
tain the following system of equations which, in combination with eqs«(4-42)
and (4»4-3), permits determining all values of Uj (see Table 1«3)
Table 1.3
do
"(
"2
«J
• • •
^Z-1
"z
"<7
bo
___
Oo
h
fl/
bi
0,
*»
a2
h
Oz
h
«j
h
Dj
• • •
tt • •
• • •
• • •
bz-Z
^2-/
bz-,
nz-,
h-,
<^i
H
Here, we have introduced the follov/ing notations:
^k.hh)
(4.45)
In this formula, at i = -1, it is necessary to substitute Pq for the values of
M_3^ and to consider the value of d,^^ as equal to unity (d^^ = 1).
With the condition (4*42), the solution of the system (4»44) reduces to
determining the values of Uj from sinple recurrence relations of the type of
*/-!
[^/-l — ^-2«/-2 — ^/-l«/-l] •
(4-46)
On solving the system (4 •44) with the condition (4*43)^? the values of Uj 749
can be represented as
«i = «0+Mi,
(4.47)
where Uq = 0, and we can determine ui from eqs.(4*46), after which the value of
Uq can be determined by means of the formula
tfo^
(4.48)
The further course of successive approximations has been described above.
49
llillll IF
In the exaniined case of hinged blade attachment at the root, the method of
successive approximations leads at first to a determination of the mode of the
fundamental, which, when the blade hinge coincides with the axis of rotation of
the rotor, will coincide with a straight line# It is natural that, in this
particular case, the calculations should "begin directly with determination of
the first harmonic, carrying out in each approximation orthogonalization to the
fundamental which will "be assumed as coinciding with a straight line.
In most designs, the root hinge of a helicopter "blade is set off from the
axis of rotation of the rotor by some amount ro, which may be as much as 3 - 10^
of the blade radius. The presence of this offset causes the mode of the funda-
mental of a hinge-suspended blade to deviate slightly from a straight line and
the natural frequency to differ noticeably from an amount equal to the rpm of
the rotor. To illustrate this effect, we will present (see Fig. 1.14) a graph
of the mode of the fundamental, for a large offset of the axis of rotor rotation
from the root hinge.
?• Case of a Blade Rigidly Attached at the Hoot
The calculation of the natural vibration mode for a blade rigidly fixed at
the root differs little from the above case of hinged attachment.
The first stage of the calculation, involving a determination of the bend-
ing moments Mj, is carried out in the same manner as described above, except
that we now solve the system outlined by a broken line in the table of eq.(4*35)*
This system incorporates one more equation in which, by virtue of the boundary
conditions, we set p = 0.
This condition is used also in solving the system (4*44), in which the co-
efficient Do is calculated from the formula
Do = CoMo + doMi,
8. Possible Simplifications in Calculating the Coefficients
We would like to enphasize that, in cases in which the blade is divided
into a sufficiently large number of segments so that the value of the coeffi-
cients a^ in eqs.(4.20) is less than 0.05 - 0.08, it is possible to sinplify
eqs.(4-2l) and (4*22) on replacing their hyperbolic functions by the first terms
of their e^cpansion in series.
Actually, in eqs.(4*2l) and (4*22) we set
sinha=a^ ^» . . ^ a-4 ;
' 31 ^ 51 ~ ^ 6 '
3 "l5 3
and neglect the values a^ with respect to unity. Then the coefficients di /50
50
and ej can be calculated from the approximate formulas
dr-
QE/,
i,i + l
h.
hJ±^ = 2di.
3E/u
+1
These siirplifications render the calculation somewhat less laborious, which
is inportant when using manual means •
9'. Certain__ Results of Calculating the Natural Blade
Vibration_ Modes and Frecaiencies
Here we distinguish two problems which are of prfane interest from our point
of view.
The first problem concerns the refinements yielding final results for the
calculation of natural blade vibration
frequencies and modes in a centrifugal
force field, in conparison with the
approximate method of calculations
presented in Section 3* After this,
we will give a discussion of the occur-
rence of sharp bends in the blade under
the effect of local phenomena of the
distribution of rigidity and mass para-
meters over the blade length. The oc-
currence of such flexures is charac-
teristic for beams extended by centri-
fugal forces and is never observed in
the absence of extension by centrifugal
forces .
TABLE 1.4
Harmonic of
Vibrations
Natural Vibration
Frequency
j^proximate
Method
I^xact
Method
Hinged suspension
at blade root
First
Second
Third
Rigid attachment
at blade root
First
Second
Third
405.3
708.5
1069.7
212.1
463.7
82U5
404.3
705.9
1069.0
194.7
461.9
817.5
Let us begin with the first prob-
lem: We already noted in Subsection 1
of Section 3 that the approximate
method of calculation of natural blade
frequencies in a centrifugal force
field, which is based on the assunption
that the natural vibration modes do not
differ in the presence or absence of
centrifugal forces, yields conpletely satisfactory results at these frequency
values .
To confirm this assuirption, let us present the values of the natural vibra-
tion frequencies of the first three harmonics of hinged and rigid helicopter
blades in a centrifugal force field • The values of the frequencies calculated
^j the approximate energy method (see Sect ,3) are shown in the second column of
Table 1.4# For conrparison, the third column contains the exact values of fre-
quencies calculated by the method presented in this Section,
51
A conparison of the frequency values presented in Table 1.4 shows that, at
hinged suspension of the "blade, the difference in their values is quite small.
At rigid attachment, the difference is somewhat greater hut still moderate.
Therefore, as pointed out above, in calculations with the purpose of preventing
the possible occurrence of resonance, the approximate method gives satisfactory-
results •
Centrifugal forces have a stronger effect on the natural vibration modes
and especially on the distribution of bending moments and curvature of the
elastic Une over the blade length.
I2k
WZtrm
RLgiare l.S shows hinged modes of the
first five harmonics (excluding the funda-
mental) for the same blade as in
Table 1.4, while Pig. 1.9 gives the distri-
bution of bending moments corresponding
to these modes • The solid Unes in
Figs. 1.8 and 1.9 (just as in Figs. 1.10,
1.11, and 1.12) represent the natural vi-
bration modes in a centrifugal force field,
and the broken Hnes indicate the same
modes for a nonrotating blade.
Figure 1.10 shows the natural vibra-
tion modes and the corresponding bending
moments for the first two harmonics of" a
blade fixed at the root.
As indicated ^s^ all these graphs,
consideration of centrifugal forces, in
certain blade sections, has a noticeable
effect on the natural vibration mode, a
point especially manifest in bending
moment diagrams and hence in the distribu-
tion of bending stresses over the blade
length. This effect is stronger, the
lower the harmonics of natural vibration.
Fig. 1.8 Modes of the First Five
Harmonics of a Blade in a Centri-
fugal Force Field and at n = 0.
The distribution of bending moments
over the blade length during its vibration
in a centrifugal force field is character-
ized by an increase in bending moments in
certain blade segments due to their de-
crease in adjacent segments. We will call this local increase in bending moments
a "concentration of bending moments". The occurrence of such concentrated bend-
ing moments is associated with the presence of large concentrated loads and
marked decreases in flexural rigidity in the blade structure.
Concentrated bending moments lead to an intensification of bending stresses
at various blade segments, caused by sharp flexures of the blade at these
segments.
52
This is of considera'ble interest for practice and thus should be studied
in greater detail*
The natiire of "blade vilDrations
texTnined by the correlation between
-1000Q
Fig*1.9 Distribution of Bending
Moments over a Blade Vibrating
with Respect to the Modes of the
First Five Harmonics in a Cen-
trifugal Force Field and when
n = 0.
in a centrifugal force field is largely de-
the magnitudes of elastic and centrifugal
forces • If the flexiu?al rigidity of the
blade is sufficiently great (as is often
the case, especially in the plane of ro-
tation of the rotor; and if the centri-
fugal forces are insignificant (low ipm of
rotor), then the vibration mode will dif-
fer little from that of a nonrotating
blade.
If, on the other hand, the flexural
rigidity of the blade is low and the
centrifugal forces are appreciable, then
the form of blade deformation is deter- /52
mined mainly by inertia and centrifugal
forces and depends little on the elastic
properties of the blade. In this case,
the form of blade deformation during vi-
bration differs little from the fonii of
deformation of an ideal flexible heavy
string stretched by centrifugal forces.
This phenomenon is generally observed
during vibrations in the thrust plane of
blades in modern helicopters.
Quantitatively, the relation between
elastic and centrifugal forces can be
estimated from the coefficient c^ which
represents the ratio of the elastic po-
tential energy to the potential energy
accumulated by the blade due to bending
in the centrifiogal force field:
'AT
The values of C^j and Cn are described
in eqs.(3.3) and {3,h).
When cy > 1, the effect of the elastic
properties of the blade is greater than
the effect of centrifugal forces. When
cv < 1, the opposite is observed.
Table 1.5 gives the values of the coefficients a for a hinged blade whose
modes of operation are shown in Rigs. 1.8 and 1.9- This blade can be regarded
as a typical helicopter blade.
53
The values of the coefficients a given in TalDle 1.5 confirm the assimption
that the helicopter "blade, with respect to its characteristics in the flapping
plane, approximates an ideal f lexilDle heavy string extended by centrifugal
forces, for which a = 0.
The properties of a "blade and of an elastic string draw closer together,
the lower the overtone of the natural vibrations.
A basic featiire of an extended ideal elastic string is that its axis under-
goes sharp bends at the points of
M;
dOOO
2000
WOO
Mr
WOO
'1000
-ZOOG
-zooo
-^000
'5000
\
n
F:
■
~
\
1 *"'' "—-
V
1
\
n=165rpm
-^
■
L<1
f
f^ ■ -
n =165 rpm
A
r/1
\
)
\
\
\
y
/
vr
/
X
/
^
^\
^'
--
V
\ ,^
—
~lu
Y"
■^
"^r<
T*-
^^
'1
iir:
n:
[^
.^3:;
L;
^!_L
__
9 rm
—
—
9^>
„,;
r- +rxn
.
1 1 1 M
/
yz
n^ldS rprrr
/
J
y
-
-d
:^J .
f
>-t^
^
jn
==1
Tm
t
k^
L*
s
6
7
^ey
9
/
■^
■^^
^e-
^
-^<
t;
\
"
-—
—
H
4
'7
\
n-O
r*
T
/
n'O
~
/
1
1 1
h
n=!SS rpm
1
i n 1 1
1.0
0.5
f.O
0,5
Fig. 1*10 Modes of first and Second
Overtones of Natural Vibrations of a
Rigid Blade.
application of concentrated lateral
forces and at sites where the
string makes contact with rigid
elements . Such a sharp flexure
generally occurs at the site where
the string is embedded or clanped.
If a rigid segment is inserted into
the string, sharp bends will form
along the edges of this segment.
Therefore, in cases when the prop-
erties of the blade and those of
the stretched string approach more
closely, the same characteristics
become manifest also in deforma-
tions of the blade. Of course, an
elastic blade, no matter how low
its flexural rigidity might be,
cannot undergo such sharp bends.
Nevertheless, sharp bends inherent
to an ideal elastic string are
transmitted to the blade and /53
cause sharp alternating bendings
of its axis. These bends are ac-
conpanied by concentrations of
bending moments and an increase in
bending stresses at the points of
flexure .
Let us examine several exairples
that confirm this assunption.
Plgure 1.11 shows the distribi>-
tion of bending moments over the blade length, corresponding to the natural vi-
bration modes of the first and second harmonics with a load almost equal to the
weight of the blade and located at a relative radiios r = O.Z^S.
At the point of attachment of the load, there is a marked concentration of
bending moments leading to an increase in stresses by a factor of almost 2 in
conparison with a nonrotating blade. The introduction into the blade of a seg-
ment of high rigidity leads to a concentration of bending moments in the area
of this segment (Pig. 1.12). However, since an increase in flexural rigidity
leads to an increase in the moment of resistance over the length of the rigid
54
TABLE 1.5
Harmonic of
Vibrations
First
Second
Third
Fourth
Coefficient a for
Deformation
in Flapping
Plane
Deformation
in Plane
of Botation
segment, the greatest stresses "will
arise along the edges of the segment,
i.e., where the ideal rigid string
would undergo sharp bends.
I3k
0.083
0.332
0.629
1.116
2.2
3.7
7.7
The occurrence of the same proper-
ties of an ideal elastic stretched
string eijq^lains the occurrence of sharp
concentrations of bending moments in
the case of rigid "blade attachment,
since a flexible string would have, at
the site of attachment, the same sharp
bend as a hinged blade.
The bending moment corresponding
to the first harmonic in the case of a rigid blade rises by a factor of almost 6
(see Fig.l.lO) in comparison with the moment of a nonrotating blade. Such a
sharp concentration of bending moments has a noticeable effect even on the
values of the natural vibration fre-
quencies (see Table l.Zf) . This
greatly reduces the feasibility of an
approximate method (see Sect .3) > as
appHed to a calculation of a blade
with rigid attachment at the root.
In practice it is often necessary
to introduce additional hinges into
the rotor blade or to shift the posi-
tion of the hinges already present in
the hub design. The necessity of
providing additional hinges has to
do with the need to reduce the bend-
ing stresses at some blade segment
or with the change in its natural
vibration frequency.
Let us now investigate the
manner in which bending deformations
of a blade are affected by the intro-
duction of an additional hinge. It
was mentioned earlier that the blade
of a helicopter is close in its
characteristics to a stretched elastic
string. A stretched chain, with
hinges continuously distributed over
its length, behaves like an elastic
string. Therefore, we can assimae
that a helicopter blade takes ap-
proximately the same shape as an ex-
tended multihinged chain during de-
formation. Thus, it is logical that
the introduction of an additional
-mn
Fig. 1.11 Bending Moment during Vibra-
tions with Respect to the First and
Second Overtone, for a Load at Radius
r = 0.48 Close in Weight to the Weight
of the Blade.
55
Illllllll
M
/55
0,1 0,2 0,3 a* 0,5 0,6 OJ 9.8 0.3 P
Fig. 1.12 Mode of Bending Moment with Respect to the First
Harmonic, for a Blade with a Segment of High Rigidity.
Rig. 1.13 Mode of First Harmonic of Natural Vibrations of
a Blade with and without Additional Hinge.
a and "b - Modes of first harmonic in a centrifugal force field
without hinge (a) and with hinge (h); c - Mode of first harmonic
of nonrotating "blades with hinge; d and e - Modes of iDending
moment with respect to first harmonic in a centrifugal force
field without hinge (d) and with hinge (e).
56
hinge into the blade cannot substantially affect the mode of its deformation.
This is illustrated in Eig.1,13 which gives the mode of the first harmonic of
natural vibrations of a blade with and without an additional hinge* It is also
seen from Fig •1-13 that the addition of an auxiliary hinge has a noticeable
effect on the mode of the bending
yt — ] i i ] I I I ^ moment only in a small segment
close to the hinge. Its influence
is negligible in segments remote
from the hinge.
Q.S
as
Ck
az
f\
1 ~-
y
-
L-**^
}0rt
,<"
y
~y,
x
^
y^ '
Hinge axis-^
1 . I
:^
y-
" —
^
at 0,1 oj a* as o.6 oj oj 0.9
Fig. 1.14 Modes of Lower Harmonic of
Natural Vibrations of a Blade with Hinge
Set Off from the Axis of Rotation and the
Bending Moment Corresponding to this
Mode (on Vibration in the Flapping Plane
p^/n = 1.35, on Vibration in the Plane
of Rotation p^/n = O.9I).
It is especially necessary
to note that, in the case in
question in which the blade has
two hinges, its vibration modes /56
in a centriftigal force field
differ greatly from the oscilla-
tion modes of a nonrotating blade.
The nonrotating blade is not de-
formed at all in first-harmonic
vibrations. Therefore, in the
given case the approximate energy
method of frequency calculation,
in the form in which it is pre-
sented in Section 3, is not appli-
cable •
Nor can we disregard the centrifugal force field in studjrLng the blade de-
formations in a Dorschmidt-type rotor with a hinge far removed from the axis of
rotation. The vibration mode of the lower harmonic of the blade of this rotor
and the corresponding bending moment are shown in Fig.l.l^. Mthout considera-
tion of centrifugal forces the mode of the blade would coincide with a straight
line and it would be inpossible to find the magnitude of the bending moment
plotted in Fig. 1.14, which is very great for this rotor and actually determines
the possibility of its use.
These exajiples show that, in many cases, the natural vibration modes in a
centrifugal force field substantially differ from the corresponding modes of a
nonrotating blade. This must be taken into account when designing a blade.
Therefore, in the design office, if the calculations are all carried out on
electronic coirputers and the degree of conplexity of the method is of no iirport,
there is no sense in resorting to approximate methods.
Section 5» Torsional Vibrations of a Blade
1. I^oblems Solved in Calculat ing Torsional Vibrations
It was noted above in Sections 1 and 4 that the calculation of the modes
and frequencies of natural flexural blade vibrations not only has a secondary
value (for stress analysis) but also an independent value as a method for select-
ing blade parameters that prevent the occurrence of bending resonance. This
problem does not arise in calculating free torsional vibrations since vibrations
57
of noticeable anplitude caused "by torsional resonance are never encountered in
practice • As a rule, apprecialDle torsional vilDrations are set vp only during
flutter or during forced vi"brations under conditions close to flutter. There-
fore, the magnitude of the frequency of natural torsional vi'brations is of no
practical interest in itself (if we do not regard it as a parameter character-
izing the torsional rigidity of a "blade), and the results of the calculation /57
of natural vibration modes and frequencies are only of secondary significance
for calculating flutter or for calculating "bending stresses coirputed with con-
sideration of torsional "blade deformations. The other problem does not arise
when calculating free torsional blade vibrations.
Two main problems are enco-untered in calculating forced torsional vibra-
tions* The first is the determination of elastic blade deformations whose con-
sideration is necessary for the calculation of bending stresses; the second is
the determination of the magnitudes of the hinge moments necessary for calcu-
lating the rotor control system.
2. Differential Ecniation of Torsional Blade yibrations
let us represent a blade in the form of a cantilever straight bar with a
torsional rigidity GT^ variable over its length. The mass moment of inertia of
the bar sections relative to its axis Im will be assimied, just as the torsional
rigidity, to be a continuous function variable over the length of the bar, the
centers of gravity of all sections of the bar to lie on its axis, and the mount-
ing of the bar to be torsionally elastic.
It is logical that reducing the problem of blade vibrations to calculation
of such a model presipposes the use of numerous simplifying assuirptions. Let
us assume that the flexural axis of the blade is rectilinear and coincides with
the axis of the feathering (axial) hinge of the rotor hub. Let us equate the
flapping corrpensator nto zero.
Allowance for displacement of the centers of gravity and determination of
the effect of the flapping conpensator on the natural frequencies will be ex-
amined in Section 6.
Use of the above assuirptions permits solving the problem of torsional blade
vibrations conpletely independently, without relating them with the flexural
blade vibrations.
Let us construct the differential equation of torsional blade vibrations.
The torque in the blade section can be determined from the differential equation:
where 3K is the linear torque of external and inertia forces acting on a blade
element.
Under the effect of torque, each element of the blade is twisted through
an angle of
58
^9= — — dr,
^^^ (5.2)
where cp is the elastic angle of rotation of the iDlade section.
The value of the torque, derived from eq.(5*2), is substituted into eq.(5«l).
Then, the differential equation of torsional deformations of the tlade can be
written in the form
Let us examine the torsional vibrations of a rotor blade rotating in a
vacuum. The linear torque in this case will be equal to
^=-{,n9-^HIy-i;i9^ (5.4)
where ly and I^ are the mass moments of inertia of the blade section relative
to its principal axes of inertia.
If the length of the profile along the x-axis is appreciably greater /5B
than along the y-axis (and this is usually the case), then we can set approxi-
mately
ly-h^U (5.5)
where 1^^ is the linear mass moment of inertia of the blade section relative to
an axis going through its flexural axis.
After substituting eq.(5*4), with consideration of eq.(5.5), into eq.(5.3),
we obtain the differential equation of torsional vibrations of a rotor blade
rotating in a centrifugal force field:
[G'^t <pT-/„(<p+«>^<p)=0. (5.6)
The blade model discussed here has the following boundary conditions:
at r=0:
at r=R: (5.7)
[GT, «>']^=0, J
where
Coon ~ rigidity of the rotor control system reduced to the axial hinge
of the hub (the control rigidity determines the magnitude of
rigidity of the elastic blade attachment at the root);
cpo = rotation of the blade in the axial (feathering) hinge as a conse-
quence of deformation of the rotor control system.
59
3. Determination of the Natiiral Torsional Blade
Vitiration Modes and Frequencies
Here, we will use the method of solution presented in Subsection 1 of Sec-
tion 2 for determining the flexural vibration modes and frequencies. let us pose
<p(/f)=(psinv^. (5.8)
Substituting eq.(5»8) into eq.(5»6), we obtain
[CT^f ?']' + (v2-^^)/„.?=0. (5-9)
It immediately follows from this equation that the natural torsional vibra-
tion modes of a rotating and nonrotating blade are identical and that the fre-
quencies are correlated by a sinple relation of the form
v2=v2 -1-0)2, (5.10)
where
V = natural frequency in a centrifugal force field;
Vq = natural frequency of blade of a nonrotating rotor.
Integrating eq*(5.9), with consideration of the boundary conditions (5.?)
for the case cjd = will yield
cp=v2
^■irl'-'^'+itS'-'"'
Cfoft
(5.11)
Here and below, we will omit the siperscript of v, which denotes that the
natural frequency is determined for u) = 0.
Equation (5. 11) is solved by the method of successive approximations, just
as had been done in solving eq.(2.4) in Section 2.
Let us prescribe an arbitrary vibration mode cp. This mode should be /59
normed in some manner, for exanple
T/?=l. (5.12)
where cpf, is the elastic angle of twist of the blade tip.
Then, performing the operations prescribed by eq.(5»ll), we determine the
function
f R R
Or
We can determine the natural torsional blade vibration frequency from the
60
noiming condition for eq.(5-l2)
««' (5.14)
where ^^ is the value of the fimction t^ at r = R.
After prescri^bing a new value of the function
?=v2& (5.15)
and performing the operations (5*13) and (5*14) as many times as necessary for
securing the required accuracy, we obtain the final values of v and cp# As in
the determination of the modes and frequencies of natural flexural vibrations,
this method of successive approximations leads to determination of the lower
harmonic of natural torsional vibrations • When determining the next harmonics,
the condition of orthogonality
J/^cp(^)<p(-)^r-0. (5^16)
must be satisfied.
Here the index j denotes the mode of the unknown harmonic of vibrations,
while the index m gives the modes of already determined lower harmonics.
Putting
^(/)==v2U^ 2 ^-^^"^ ' (5.17)
we obtain from the condition (5.16) the following expressions for the constant
coefficients c^ :
The natural vibration frequencies of subsequent harmonics are determined
in each approximation from the formula
1
*/?- 2 'm (5.19)
m«l
Upon conpleting the determination of all natural vibration modes and fre-
quencies needed for fxorther calculations it is necessary to correct the frequen-
cies by means of eq.(5»lO) which takes into account the effect of centrifugal
forces •
61
Calculations of the natural torsional vibration modes and frequencies of a
"blade in actual helicopters show that the rigidity of the rotor control system
is of decisive inportance in determining the magnitudes of lower-harmonic /60
vilDration frequencies • The torsional blade rigidity is almost always much
higher than the rigidity of the control system. RLgure 1.15 shows the modes of
the first harmonic of natural torsional blade vibrations for different heli-
copters in mass service.
Based on the relationship between the torsional deformations of the blade
and rotor control system in first-hax^nonic vibrations, it is possible to esti-
mate the extent of torsional rigidity of the blade in coirparison with the
rigidity of the control system. The correlation between these rigidities is
estimated by the coefficient a (see Fig.1.15). This coefficient determines the
portion of the total angle of rotation of the blade tip due to deformations of
only the blade.
to
0,8
0,6
OM
0.2
—
:==
r^
Is
\
^_^^-»— ^
i «
-^
1 k h "^
0.1 O.l 0,3 OA 0,5 0,6 0,7 0,B 0,9 tO
Fig.1.15 Natural Torsional Blade Vibration Modes; for Various
Correlations of Blade and Control System Rigidities.
The described characteristic in the correlation between blade and control
rigidities permits in certain calculations the assunption that the torsional
blade deformations are small in comparison with the control deformations, thus
making it possible to use only the blade twist due to deformation of the control.
This assuirption is often used in the calculation of flutter (see Chapt.IV of
Vol.1).
The results of the calculation by the above method pennit an estimate of
the type of layout of the natural torsional vibration frequencies of a blade
relative to the harmonic conponents of aerodynamic forces. Figure 1.16 gives
the resonance diagram of torsional vibrations of a blade constructed for one of
the existing helicopters, while Fig. 1.17 shows the modes of the first three
harmonics.
It was noted in Subsection 1 of this Section that the variable external
forces producing blade twist are small so that, even in the presence of resonance,
the torsional vibration anpHtudes do not become dangerous for the strength of
62
35 33 31 13
M
Fig. 1.16 Resonance Diagrani of Torsional Blade Vibrations-
7.0
0.8
as
OM
02
'0.Z
-DM
-0,6
-as
-to
.9"^
^
r
\
/
/
\
-{pi^}
/
/
/
\
9^
/
/
/
/
/
'
0.
1 a
z 0,:
V
'^ 0.
./.,
6 a
7 0.
1
s a
9 r
\
/
/
./
\j
_^
^
\ -
Fag. 1.17 Modes of I^rst Three Harmonics of
Torsional Blade Vibrations.
63
the "blade • In view of this^ one usually does not try to avoid torsional reso-
nance, and the resonance diagram presented in RLg#l#l6 is given only for esti-
mating the alDsolute magnitude of the torsional vibration freqaencies.
It follows from Fig»l«16 that even the second harmonic of torsional vibra-
tions at the operating ipm n^p proves to be higher than the 15'^'^ harmonic of the
rotor rpm. The freqaencies of subsequent overtones are even higher. Therefore,
probably only the frequency of the first harmonic of natural torsional blade
vibrations can be of practical interest.
All of the above considerations pertain to torsional vibrations of a rotor
blade treated as an isolated blade, without consideration of the relations 762
sxjperiirposed on the vibrations by the design of blade attachment at the hub.
It was found that the interconnection of torsional vibrations of individual
rotor blades across the control system may substantially change the entire pat-
tern of vibrations.
4. Determination of the Natural Vibration Modes and
Freqaencies of a Rotor as a Whole
I^gure 1.18 gives the diagram of the blade-setting control system used on
most modern helicopters. Designwise this system is laid out so that loading of
one or another control loop depends
on the combination of forces generated
at the swaslplate of the pitch control
by the blades . The form of this com^
bination depends on the vibration
mode of the rotor, i.e., on the dis-
tribution of vibration phases with
respect to the blades. For exairple,
when all blades vibrate with the same
phase, the control loop is loaded only
by the total pitch. When opposite
blades vibrate in opposite phase, the
lateral and longitudinal control loops
are loaded. Finally, if the n-umber
of rotor blades is more than three,
vibration modes become possible at
which all forces arriving from the
blades are locked on the swashplate
of the pitch control.
Fig. 1.18 Diagram of Pitch Control.
1 - Blade turning lever; 2 - Flap-
ping hinge; 3 - Drag hinge;
4 " Blade; 5 - Swashplate of control;
6 - SHde.
Variable forces during vibrations
cause deformations of the control
loops loaded by these forces. On
deformation of individual control loops, the swaslrplate of the pitch control is
set in oscillation; these vibrations inpose definite phases on the blade vibra-
tions. For exairple, during vertical vibrations of the swaslplate generated by
deformation of the collective pitch control loop, rotor vibrations are excited
having a mode in which the phases of all blades are identical.
64
II II 1 1
I I
■ III II uiiiiui r II
If the swastplate of the pitch control is inclined during vibration, oppo-
site blades are excited in opposite phase. Thus, the swaslplate of the pitch
control coiples the vibrations of indi-vddual rotor blades. As a result, blade
vibrations can occur only with well-defined vibration modes of the entire rotor
as a whole, and the nijmber of such modes will coincide with the number of rotor
blades • Here, each vibration mode corresponds to its value of control rigidity-
reduced to the feathering hinge of the blade, which depends on the rigidity of
the control' loop loaded at this mode. Accordingly, each mode of rotor vibration
is characterized by its own frequency value of the natural torsional blade vi-
brations .
Consequently, for a rotor with a number of blades z^j there are z^ different
natiiral vibration frequencies corresponding to each harmonic of torsional /63
blade vibration. Each natural vibration frequency is characterized by its own
specific mode of distribution of angles of twist over the blade length, but
qualitatively all modes corresponding to a specific harmonic of vibrations do
not differ; for exairple, they have an identical nimiber of vibration nodes.
As a typical exanple we can cite the values of the natioral vibration fre-
quencies of the first harmonic for the four-blade rotor of the Mi-4 helicopter.
The lower frequency values at stressing of the longitudinal and lateral
controls, relative to the operating rpm of the rotor, are --^ — = 3*4 to 3*5»
"op
L^on loading the collective pitch control, this quantity teikes the value
— \^*^ - 4*6 while, upon locking all forces from the rotor on the swaslrplate,
we have ° * ^ =6.6*
A very iirportant circumstance is that only the first harmonic of natural
torsional blade vibrations lies within the limits of the vibration frequencies
corresponding to harmonics of the rotor rpm, with respect to which the external
forces have a noticeable magnitude. All subsequent harmonics of ' vibrations lie
higher and therefore are of no practical interest .
Section 6. Combined Flexural and Torsional Blade Vibrations
1» Coupling of Flexural and Torsional Vibrations
Above, we discussed free flexural and torsional blade vibrations as two
unrelated, independent problems. In a real blade, torsional and flexural vibra-
tions are always related. The intensity of such coipling will be demonstrated
below. We will examine blade vibrations in vacuum, when the coupling between
torsional and flexural vibrations is produced exclusively by displacement of
the centers of gravity of the sections relative to the flexural axis of the
blade and as a consequence of the kinematic coij^jling over the flapping compen-
sator. We will use the method of calculation constructed on the basis of the
three-moment method described in Section 4, as applied to calculation of flexural
vibrations .
65
The possi"bility ot calculating the natural flexural and torsional ("binary)
vi"bration frequencies is useful to the designer in solving numerous specific
practical prolDlems.
For example, such a calculation becomes necessary if it is desired to place
outrigger telancers on the "blade to prevent resonance • Here we have in mind the
relatively rare cases when the use of "balancers is proposed not to eliminate
flutter "but to change the natural frequencies.
The designer may wish to take into account the coipling "between flexural
and torsional vi"brations also when the calculation of natioral blade frequencies
for some reason does not coincide with experiment. Here it can he shown in many
cases that this difference is due to disregard of such coiplLng. We can hope
that the calculation results presented "below will facilitate settling these
douhts .
We should note, however, that calculation of natioral frequencies in /6k
vacuum cannot give the answer to many questions in practical use having to do
with the appearance of high variable stresses of some frequency in the blade,
which are evaluated as resonance since aerodynamic forces often introduce sub-
stantial corrections into the picture of the phenomenon.
2. Method of Calculating Binary Vibrations
Calculation of natural binary vibration modes and frequencies is greatly
sinpHfied if we only, consider blades of a definite conventional type, for
whose calculation the following assunptions can be used:
1. The flexural axis of the blade is a straight line coinciding with the
axis of the feathering (axial) hinge.
<:. ■
Uis Of f, 3,
Flexural axis
Center of gravity
— --kjU —
Fig. 1.19 Design Model of Blade.
66
The method of calculation does not fundamentally change when these axes do
not coincide. It is then only necessary to introduce, into the calculation
formulas, a number of additional terms which take into account the distance be-
tween these axes. For simplicity of confutation, we will assume that the
flexural axis goes through the axis of rotation of the rotor.
£• The plane of minimum blade rigidity is considered to coincide with the
flapping plane.
3» The "blade performs torsional vibrations as a consequence of torsional
deformations of the blade itself, deformation of the pitch control, and kine-
matic coopUng over the flapping conpensator with blade vibrations in the flap-
ping plane.
These assurr^^tions pemiit representing the blade as a weightless free beam
divided into z segments, along whose edges loads are placed of a mass m^ at
some stagger x^.g (Fig. 1.19). Each load, in addition to the mass m^ concen-
trated at the center of gravity of the corresponding blade element, has a
certain moment of inertia I^.g relative to an axis going through the center of
gravity of the load and parallel to the elastic axis of the blade.
let us represent the flexural and torsional rigidities in the form of
stepped curves, such that they remain constant over the length of each segment.
The presence of a flapping conpensator leads to a kinematic coupling /65
between the flexural and torsional vibrations, which can be expressed by the
formula
M,
Ocon
(6.1)
where
cpt, = angle of blade rotation in the feathering hinge;
M* = twisting moment relative to the feathering hinge;
Ccon = rigidity of the pitch control reduced to the feathering hinge;
H = flapping compensator;
Po ^ angle of blade rotation relative to the flapping hinge.
Furthermore, the boundary conditions at the root of the hinged blade some-
what change during its vibrations in the thrust plane. These conditions, in
the presence of a flapping conpensator, can be written as
M^^'AMto. (6.2)
where Mq is the bending moment and M + ^ is the twisting moment at the blade root.
In constructing the differential equations of blade vibrations in the flap-
ping plane, we will use the three-moment method in the form as presented in
Section 4» ^plication of this method to the case examined here leads to the
following equations:
67
Here,
z
Qi-i. i = — 2 '"i^i (wherei^O, l,2,...,z);
(6.4)
where
fj - vertical displacement of points of the elastic "blade axis (see
Fig. 1,19);
yj = vertical displacement of the centers of gravity of the loads mj.
The expressions for the constants a^, iDi, c^ , hj, and gi are given in
Section 4; see eqs.(4»lS) - (4»25)»
The displacements of the elastic axis fj and the centers of gravity of the
tlade elements y^ are related ty
/i=^i+-^c.yi?/. (6.5)
where cpi is the angle of rotation of the "blade elements about its elastic axis.
To determine the "binary vibration modes and frequencies of a blade,
eqs.(6.3) must "be sipplemented "by the equations of torsional vibrations.
The twisting moment, if it is considered constant in magnitude over the
length of each blade segment, can be defined as
M
^ i-^.r "2 ^c, ,9i -^'Ii^c.,,^i+^'y,m,x,,^^rJ'^+ ^rn,x,^uyi' (6.6)
From the magnitude of the twisting moment, we can determine the torsional /66
deformations of the blade
^^-^.^^-'-^
t'O'
G^*,_, ,. (6.7)
where GT^ is the torsional rigidity of a blade segment having a length equal
1 ~ X^l
■*^o li-.i,i while cfb is determined by eq.(6.1).
When using the three-moment method, the boundary conditions of the problem
are taken into account in the coefficients of the equation of the system. Thus,
in the case examined here, the boundary condition (6.2) leads to a change of the
coefficients of the first two equations of the system (6.3). For a blade with
hinged attachment at the root these equations" can be written in the following
manner :
68
first equation of the system (6#3)> from which the value Po is deter-
mined:
?o + -^go^t, -h,M, = ^; (6.8)
second equation of the system (6.3) J
■-^o^^o+^i^i+^i^2 = ^-^. (6.9)
Thus, the system of equations that includes eqs.(6.3), (6.5), (6.6), and
(6.7) represents a system of differential equations of binary "blade vibrations.
The solution of this system pennits determining the modes and frequencies of
natural binary blade vibrations, which also enters into the calculation problem.
If we assume that the variables entering the differential equations (6.3),
(6.5), (6.6), and (6.7) vary in accordance with a sine law of the type
then these equations can be transformed into a system of algebraic _ equations
relative to unknowns representing the anplitude values of the previoios variables.
The parameters p^ and y = — — -will enter as cofactors only into certain coeffi-
cients of these equations. If we set p^ = 1, then these equations can be re-
vrritten in the form
h:_,Mt_, + g,M, + h,JV!.^,-.
.Qij-
Z Z z
/
where
^i = ^i~^c.g,^i^ (6.13)
z
¥,■
69
The quantities entering these equations obey the follovdng relations:
fi = P%\
^ l-Ul ^ f i~Ui
% = P'?0^
(6.14)
The system of equations (6.10), (6*11), (6.12), and (6.13) can he con^
viently solved by the method of successive approximations. In so doing, in each
approxijnation we should refine the parameter y for the angular velocity of ro-
tation of the rotor u) prescribed in the calculation •
The successive approximations are carried out in the following sequence:
Assign a certain magnitude of the parameter y and an arbitrary form of the
zeroth approximation of the functions yj and cpi .
Normalize the functions taken as the zeroth approximation, for exanple
After this, derive the function f^ from eq.(6,5). Then, from eq.(6.1l),
detennine the quantity M^ needed for solving the system of equations (6.10).
At the same time M^, ^ , is determined.
1 ~ J. ji
After solving the system of equations (6.10) and determining Uj from the
first equation of this system, determine po :
(6.15)
Then, from eq.(6.l2), determine '&^ and from eq.(6.13) the values of v^ ,
z
which, furthermore, should satisfy the condition S m^ v^ = 0.
Determine the natural frequency from the condition of normalization on the
basis of the first relation in the system (6.14), thus:
p^^.
(6.16)
After this, determine the functions yi and cpi from eq.(6.1[|-) and use them
in the next approximations which are performed in the same sequence. At the
same time, refine the parameter y
70
This method of successive approximations leads to a determination of the /68
frequency and mode of the lower harmonic of natural vibrations. To determine
the next harmonic, we use the condition of orthogonality which for binary vibra-
tions has the following form:
Here, the index j denotes the mode of the sought harmonic and the index m
the modes of already determined lower harmonics.
The use of this method of calculation gives results that are conpletely
satisfactory for practice.
It should be mentioned that, in cases in which the natural frequencies of
two successive harmonics have sufficiently close values, this method of calcula-
tion does not give a converging solution. In practice, however, this is of no
great inport since it can happen only when the coupling between torsional and
flexural vibrations is very weak and the corresponding vibration modes can be
determined separately without consideration of this coi:pling.
3* Effect of Coupling between Bending and Torsion
at Natural Vibration Fre^Cfuency
Here, we will define the extent of the difference of natural binary blade
vibration frequencies from corresponding partial frequencies, i.e., frequencies
obtained without consideration of coupling between bending and torsion.
Calculations show that the coipUng between bending and twisting has the
greatest influence on natural blade vibrations in regions in which the partial
frequencies of bending and torsion approach closely. Therefore, we should in-
vestigate only these regions. Outside these zones, the partial frequencies of
the blade and the frequencies of the coiipled binary vibrations practically
coincide .
It is known that the partial frequencies of natural vibrations of bending
of a hinged blade, for all modern helicopters, lie in very narrow well-defined
zones whose position relative to the harmonics of external excitation cannot be
changed substantially. In Fig. 1.20, these zones are superposed on a resonance
diagrajn of the blade. This diagram is constructed for the frequency range that
includes only a series of first harmonics of rotor ipm, since external forces
acting on the blade with higher harmonics are insignificant in magnitude and
cannot cause noticeable blade vibrations. Only the first three overtones of the
partial frequencies of the blade in bending fall within this region. In practice
only these overtones are of interest in blade design. The natural flexural vi-
bration frequencies can leave the indicated zones only for rotors with an iinusual
method of blade attachment to the hub, such as - for exairple - in rotors with
rigid blade attachment or with a gimbaled hub.
The partial frequencies of natural torsional blade vibrations may vary
71
within wider lindts, mainly as a consequence of the difference in the rigidities
of the rotor control system whose designs may vary widely. Nevertheless, with
respect to the magnitudes of the partial frequencies of natural torsional "blade
vibrations a very inportant concliision can be drawn, involving the following:
Only the first harmonic of torsional vibrations can fall within the frequency
range of interest here. The second harmonic of torsional vibrations generally
will be in a region not below the 15"^^ harmonic of the rotor rpm (see Fig. 1.16),
i.e., beyond the limits of the region of interest to the designer. Usually, 769
vibrations of relatively large airpHtude do not arise with such frequencies.
Therefore, only the first harmonic of natural torsional vibrations of a blade is
of practical interest from the aspect of possible occurrence of resonance.
9 th harmonic s
10th harmonic
8th harmonic
7th harmonic
6th harmonic
5th harmonic
Ath harmonic
3rd harmonic
2nd harmonic
1st harmonic
to n/npp
Fig. 1.20 Regions on the Resonance Diagram of the Frequencies
of Natioral Vibrations of the First, Second, and Third
Overtone of Bending and the First Overtone of Torsion
for Blades of Different Helicopters.
Here, it should be recalled that the helicopter rotor blade may have several
first overtones of torsional vibrations with different frequencies, depending
on the vibration mode of the rotor as a whole and on what control loop is loaded
at this mode. The difference in natural vibration frequency of these modes will
be determined exclusively by the difference in the rigidities of the control
loops being loaded.
72
In flight, each harmonic of external forces is able to excite only one
well-defined -vibration mode. Therefore, in investigating the possibility of the
occurrence of resonance it is necessary to check whether the control rigidity
adopted in the calculation corresponds to this mode, with which resonance is
possible. In this Section, we will discuss only natural vibrations of the sys-
tem. Therefore, we will not fiirther discuss this problem.
Figure 1.20 shows the region which usually comprises the frequencies of the
first harmonic of natural vibrations of a blade in torsion, for all modes of
rotor vibrations when both cyclic and collective pitch control loops are loaded.
For rotors with a blade number greater than three, a vibration mode is possible
in which all forces arriving from
Flexural axis
-1.0
Fig,1.2l Stepped Centering of Blade
with a Change of Sign at the Mode of the
First Overtone of Natural Flexural
Vibrations .
the blades lock on the swashplate
of the pitch control. The /70
control rigidity corresponding to
this mode generally is very high.
In Fig. 1.20, the ipper limit of
the region of torsional vibrations
for this case is shown by a dot-
dash line.
Let us examine the most common
case in which partial frequencies
of the first harmonic of bending
and the first harmonic of torsion
coincide in magnitude in the zone
of operating rpm of the rotor.
Let us discuss two versions of the
blade center-of-gravity distribu-
tion over its length.
In both versions, we will
assume - in conformity with the
above-adopted assunptions - that
the flexioral axis of the blade is
rectilinear and coincides with the
axis of the feathering hinge of the hub. The distance to the centers of gravity
of the sections will be reckoned from the flexural axis in percentages of the
blade chord. All investigations will be conducted applicable to a helicopter
blade with a pressed duralumin spar with a chord constant over its length. Such
a blade has roughly a constant linear weight over the length. Its chord comr-
prises about 1/20 of the rotor radius.
So that the results of the calculations will be more graphic, we will as-
sume that, ipon variations in blade centering, the mass moments of inertia of
its sections relative to an axis going through the centers of gravity do not
.change, i.e., that the position
^c.j=C0^5^
is maintained.
73
iilillililll
First, let us exaniine the case in which the centerings of the blade sec-
tions are constant over its length, i.e..
'o-r
=i££ = const,
b
where "b is the tjlade chord.
This version of the distribution of centerings is considered quite wide-
spread in practice. Fiorthermore, it permits tracing - in a very graphic form -
the effect of centering and evaluating its significance as a factor of the
coupling Ibetween flexural and torsional vibrations.
Figure 1.22 shows the resonance diagram of a blade for this case. The
solid ILnes represent the partial frequencies of bending and twisting of the /7l
blade, and the dashed lines give the frequencies of binary vibrations calculated
for a displacement of the centering relative to the flexural axis, equal to ~ "
of the blade chord. The calculations were performed for the case of h =0.
Therefore, the sign of the shift of centering is of no significance.
p cyc/min
1000
800
6th harmonic
5th harmonic
Ath harmonic
3rd harmonic
2nd harmonic
ist harmonic
ZOO n rpm
Rig. 1.22 Resonance Diagram of Blade at Displacement of
Centering Constant over the Length and Amounting
to 10^ of the Chord.
Here and below, we will intentionally examine a very wide range of varia-
tion in centerings, so as to trace its influence in a more concise form. In
practice, the design capabilities and the conditions inposed by flutter permit
74
changing of the centering only within very narrow limits. Usiially, for rotor
blades the centering varies within limits from 2D% to 25% of the blade chord
(here, values reckoned from the leading edge of the blade are given), i.e., the
entire range of variation in centering amounts to only about 5^ of the blade
chord. Thus, we can conclude from a study of Fig. 1.22 that a displacement of
centering, constant over the blade length, has only a negligible effect on the
values of natural frequencies.
In the second case examined here, the distribution of centering is selected
such that its influence is strongest during vibrations with a frequency close
to the partial bending frequency of the first harmonic. The centering is assumed
as constant over the blade length, but its sign changes at the node of the first
harmonic of the partial bending mode.
p eye/ mi n
IIGO
7th harmonic
8th harmonic \ 6th harmonic
WOO
5th harmonic
■ ' r ^ - ^^ ^tfi harmonic
•^3rd harmonic
2nd harmonic
ist harmonic
200
/I rpm
Fig. 1.23 frequencies of Natural Binary Blade Vibrations with
a Stepped Law of Change of Centering over the Blade length,
at 10 and 20^ Displacement with Respect to the Chord
from the Flexural ibcLs.
An offset centering can be created for a blade when the anti-flutter
balancer is introduced into the design not over the entire length but only over
a small segment at the blade tip. Results of the calculation of this version
of centering distribution are given in Fig. 1.23* The effect of centering is
75
rather strong in this case. Therefore, at such a distribution over the length,
the cotpling between "bending and torsion must "be taken into account when calcur-
lating the "blade.
It is also necessary to examine the effect of a concentrated load shift- /72
ing over the chord. Let us take the magnitude of the load as equal to 8^ of
the "blade weight. This prolDa"bly is the maximxjm value of a load that can actual-
ly "be attached to a "blade. The most effective site of attaching such a load
from the viewpoint of producing strong coipling factors for flexural and tor-
sional vibrations is the point of the blade where the displacements in the
thrust plane are maximum. Therefore, we will discuss the case in which the load
is attached at the "blade tip.
Figure 1.2i^ shows the results of calculation for this case. The effect of
a concentrated load on the natural vibration frequency for large offset of the
load can be considered substantial; however, the use of such a means for elimi-
nating resonance cannot be recommended to the designer. Nevertheless, the at-
tachment of a load can be regarded as a tenporary means for treating blades sub-
ject to large variable stresses due to resonance.
■p cyc/min
ilQQ
7th harmonic 6th harmonic
WOO
5th harmonic
Ath harmonic
3rd harmonic
2nd harmonic
ist harmonic
nrpm
Fig. 1.2^ Effect of a 10 and 20^ Displacement with Respect to
the Chord of a 10-kg Tip-Concentrated Load on the Magnitude of
the Natural Binary Blade Vibration I^equencies.
76
The last parameter which should be regarded as a coiopling factor between
bending and torsion is the flapping conpensator. To evaluate its effect on the
magnitude of the natural binary* vibration frequencies, we made calculations with
a flapping conpensator h = 1.0 • This is the maximum value of a flapping com-
pensator of the type ever used in practice. All the data presented above were
obtained with h = 0.
It follows from the calculations that the effect of the flapping compensa -/73
tor is negligible. However, consideration of the flapping conpensator can be
justified to some extent, since it introduces some refinement into the form of
the distribution of the bending moment at the blade root.
Section ?♦ Forced Blade Vibrations
1. Use of B.G.Galerkin^s Method for Calculating Blade Deformations .
Determination of Static Deformations of a Blade
The problem of the determination of blade deformations reduces to a solu-
tion of the above differential equation (1.9) whose derivation is given in Sec-
tion 1:
where T is a Unear external load on the blade, distributed over the radius and
varying in tnjne.
In Sections 2, 3, and k, we discussed the solution of a similar equation
for T = describing the free vibrations of a blade. Here, we will examine
forced vibrations of a blade when T is some periodic function varying with the
frequency v.
In the particular case when v = 0, the problem reduces to a determination
of the static blade deformations due to a load Tq constant in time.
The sinplest method of solving eq.(7*l) is that given by B.G.Galerkin. /Ik-
To illustrate the application of Galerkin's method to the determination of
blade deformations, let us examine the static problem, when the external load is
time-invariant. In this case, y = and eq.(7»l) can be written as
lEIy^r-Wy'Y-To, (7.2)
Let us represent the blade deformations in the form
.=2;w^^ (,.3)
where
y^ = natural vibration modes of the blade with respect to the j-th
harmonic ;
77
Cihj = Ai,
Cj-
£ny"?jdr+^N
[y'?jdr; '
A,-
= ^T,yU)dr.
6j = certain coefficients which will "be called coefficients of blade de-
formation. The coefficients of deformation in all fxirther confuta-
tions in which the Galerkin method is used will play the role of
generalized coordinates of the system.
Let us substitute eq.(7.3) into eq.(7*2), multiply all terms of the equa-
tion in turn by y^°^ , j^^^ , y^^^ , etc. and integrate them with respect to the
blade radius.
By virtue of the orthogonality of the functions y^^^ , the performed opera-
tion transforms the differential equation (7»2) into a series of independent
equations of the form
(7.4)
where
Ci=JEI[yTjdr-\-JN[y']^.dr;
(7.5)
We will designate the quantity C^ as the generalized rigidity of the blade
diiring deformation with respect to the mode of the j-th harmonic in a centrifu-
gal force field. It follows from an examination of eqs.(7»5) that the general-
ized blade rigidity Cj is equal to double the potential energy accumulated by
the blade during its elastic deformation in a centrifxigal force field with re-
spect to the normalized mode of the j-th harmonic. Let us call the quantity A^
the generalized external force deforming the blade with respect to the mode of
the j-th harmonic. The magnitude of the generalized force Aj is equal to double
the work done by the external linear forces Tq during deformation of the blade
with respect to the normalized mode of the j-th harmonic of its natural vibra-
tions.
From eq.(7.4) we can determine the coefficients of blade deformation 6j :
8/=^. (7.6)
after which eq.(7.3) will yield the mode of static blade deformation.
The more natural vibration modes are used in the calculation, the more ac-
curate can the mode of deformations be determined. However, for practical pur-
poses it is sufficient to limit ourselves to the first four harmonics of- blade
vibrations .
If the coefficients of deformation 6 j are known, then it is easy to de- /75
teinnine the bending moments and the bending stresses in the blade. These are
determined from the formulas
78
J
ii.i)
Here, M^**^ and a^^ are the modes of distrilDution of bending moments and
Trending stresses in normalized deformations of the blade with respect to the j-th
harmonic of its natural vibrations •
The quantities entering eqs.(7»7) are governed by the relations:
-U).
W '
(7.8)
where W is the moment of resistance of the blade sections.
Determination of Blade Deformations with Periodic Application
of an Eybernal load
Let us here discuss the case in which the external load varies in accord-
ance with the law:
7=rvSinv2'.
(7.9)
To solve this problem, we will again use Galerkin^s method. Representing
the blade deformations in the form of eq.(7.3), we first substitute eqs.(7.3)
and (7.9) into eq.(7»l)* multiply all terms of the obtained equations in turn by
y^^^ and integrate over the blade length. By virtue of orthogonality of the
function y^''^, we obtain a series of independent differential equations of the
form
m-^i + CyS/ = A I sin v/,
(7.10)
where
R
Aj=^T,yU>dn
(7.11)
We will designate the quantity m^ as the equivalent mass of the blade during
its vibrations with respect to the mode of the j-th harmonic. If the vibration
modes y^^^ are noniied so that yi^^ = 1, then m^ will be the equivalent mass of
the blade reduced to its tip. It also follows from the first equation of the
system (7.11) that the equivalent blade mass is equal to double its kinetic en-
ergy and that the blade elements are displaced at a rate of y
(j)
79
Illllllilll
To determine the steady motion, we pose
^J'^H^lsinxL
Substituting this e^q^ression into eq.(7*lC)) and canceling all terms of the
equation "by the quantity sin vt, we obtain
-v^'n4i/+C;-8(/) = ^/. (7.12)
according to which the value of the anplitude of blade deformation is equal to /76
^^.= r ^^ v2 V (7.13)
'''['^'^i
It is not difficult to note that the ratio Cj/m^ is equal to the natural
vibration frequency of the j-th harmonic of the blade. Actually, if we set Aj =
= in eq.(7*l2), then the value of v in this case will determine the natiu?al
frequency of the blade and can be obtained from eq.(7*l2):
y2^p2^C^lmi. (7.14)
In conformity with eq«(7*6) the ratio A^/Cj determines the magnitude of de-
formation if the load Tv were to be applied statically.
Equation (7»13) is conveniently represented in the fonn
un^iiu)^ (7.15)
where
6pi^ = coefficient determining th^ magnitude of deformation at a stati-
cally applied external load Tv; below, this coefficient will be
called the coefficient of quasi-static blade deformation;
^dyn "^ coefficient of dynamic increase in vibration anplitude.
For the case in question, we have
X^ --
1- —
(7.16)
It follows from eq.(7»l6) that during resonance, when the frequency of
forced vibrations v is equal to the frequency of the natural vibrations pj, the
coefficient of dynamic increase in amplitude becomes infinite # This result is
regular for problems in which forced vibrations without danping are examined.
In reality, a helicopter blade operating in air undergoes appreciable aero-
80
dynamic danping during vibration. Aerodjmamic damping limits the anplitude of
"blade vibrations in resonance and must be taken into account if a determination
of blade vibrations, under conditions of resonance, enters into the problem of
the calculation.
In determining the vibrations of a helicopter blade, when vibrations arise
under the effect of aerodynamic forces, it is very difficult to make a strict
separation between forces of aerodynamic danping and aerodynamic forces causing
blade vibrations. Such a separation can be made only conditionally. However,
a niamber of simplified calculation methods do use such a division. Therefore,
we will discuss this approach in some detail.
3 • Simplified App roach to Ca lculation of Forced Blade Vibrations
Let us assume that the external aerodynamic loads acting on an elastic
blade in flight can be divided into two parts: external loads acting on the
blade and forces of aerodynamic danping. We will stipulate, in first approxima-
tion, that the external loads acting on an elastic blade coincide with loads act-
ing on an ideal flexurally rigid blade. Then, for performing the calculation it
remains only to determine the forces of aerodynamic damping.
Usually the forces of aerodynamic damping are determined for a regime /77
with axial flow past the rotor, whereipon it is assumed that, in all other flight
regimes with oblique flow past the rotor, the coefficients of aerodynamic danp-
ing do not change.
In a regime with axial flow past the rotor, the force of aerodynamic danp-
ing can be determined on the basis of the following:
During vibration, the blade elements move with a velocity f. As a conse-
quence, the angles of attack of all blade elements change by the quantity
co/-
U^Don a change in the angle of attack, the blade elements become subject to
the action of additional forces of aerodynamic danping
Let us assume that the aerodynamic load T can be represented as consisting
of two components:
where
Tpig = aerodynamic load acting on a rigid blade;
"^damp = additional load due to aerodynamic danping produced during elastic
81
"blade vibrations.
Then, eq.(7*l) can be rewritten in the following form:
{EIy^r-Wi}'Y-\-my+^clQb<.ry=^%,^. (7.19)
Let us examine "blade vi"brations due to the sinusoidal conponent of the aero-
dynamic load, varying according to the law
If we represent blade deformations in the form of eq.(7.3) and apply B#G.
Galerkin^s method to eq»(7«19)j then we arrive at a system of ordinary differen-
tial equations relative to the coefficients of deformation 6j • Individual equa-
tions of this system will be correlated by terms into which the following inte-
gral enters:
R
Dlm-lbryU)yim)ar^
where y^^^ and j^^^ are the natioral vibration modes corresponding to different
harmonics (j 7^ m) .
In siH^lified methods of calculation, the integrals Dj^ are usually equated
to zero although, in margr cases, such an assunption is inpossible to justify.
If we nevertheless make use of this assunption, then application of G-aler-
kin^s method yields a series of independent differential equations of the form
where the coefficient §^ determines the magnitude of aerodynamic dairping: /IB
R
After dividing all terms of eq.(7«20) by m^ , we obtain an equation of the
form
\ + 2/i/S/ + p% = p]lf^ sin v^, ( 7 , 22)
where
mj
82
Usually, for the characteristic of the magnitude of darrping we use the rela-
tive dauping coefficient
Pi
Its magnitude, as applied to aerodynamic danping of a "blade, is calculated
by means of the formula
R
~"-^-\'yfi^,V''^^'''^''"-- (7.23)
The solution of eq»(7*22), performed in the same manner as that used above
in solving eq«(7»10), leads to the formula
where the coefficient of the dynamic increase in vibration anplitude is
[-ar-^'&)
Thus, the solution of the problem exainined here consists in determining the
quasi-static coefficients of deformation b[\^ and their subsequent multiplica-
tion by the value of the coefficient of dynamic increase in anplitude Xdyn •
Such an approach is subject to certain inaccuracies because of the arti-
ficial separation of aerodynamic forces into two con^Donents by eq.(7«18), the
inadequately founded assumption that Dj^ =0, and the approximate determination
of the coefficients of aerodynamic danping for a regime with axial flow past the
rotor. Therefore, in Sections 8 and 9 "we will present methods of calculation in
which the above assurrptions are not used.
Nevertheless, a siirpl±fied approach of this type fairly well describes the
qualitative aspect of phenomena observed during blade vibrations.
4. Amplitude Diagram of Blade Vibrations /79
As indicated above in Section 3, the resonance diagram of a blade is widely
used in evaluating the character of blade vibrations. The resonance diagram
permits estimating the extent to which the natural vibration frequencies of the
blade differ from the excitation frequencies and determining the possible hazard
of the occurrence of resonance vibrations. However, in cases in which the natu-
83
■Illlllllll
ral frequencies and the excitation frequencies do not differ greatly, it is of
interest to estimate the extent to which "blade •vi'bration anplLtudes can "be re-
duced. Such an estimate can "be made "by using the anplltude diagram of blade vi-
brations. This diagram, constructed for
a blade with ordinary mass and rigidity
characteristics, is given in Pig. 1.25.
6
First
overtone
y
- Second ~j
overtone t
/I
/ tl
|: \/
"f
rd
L
^overtone
\i
XO-
Fig .1.25 Amplitude Diagram of
Blade Vibrations.
In this diagram, the abscissa gives
the excitation frequency referred to the
angular velocity of rotation of the
rotor.
v=-
(7.25)
The ordinate gives the coefficients
of dynamic increase in vibration anpli-
tude. The diagram is constructed only
for the first three harmonics of elastic
blade vibrations, using the dan^Ding co-
efficients calculated from eq.(7.23).
5. Calculation of Vibrations at .Appl lcatiori Phase of Exter nal
Load Variable over the Blade Length
In Subsection 3 of this Section we presented formulas for the case when the
external load is represented as
This form of notation of the load is possible only if the phase of its ap-
plication over the blade length is constant. As a rule, this does not happen
during vibrations of a helicopter blade. The phase of the external load varies
over the blade length, so that the load should be represented in the form
T^iQ = T^ cos \i -{-TyS in v/,
rig
(7.26)
where the conponents of the external load T^ and Tv vary over the blade length
in accordance with different laws.
After substituting eq.(7.26) into eq.(7»19)^ using Galerkin's method, and /80
assuming that Djjj =0, we obtain
where
mjZj + S/B; -{- CyS; = .4; COS Vl -j- ^; sIh Vr.
(7.27)
84
^ ^_
Aj^^7\y<^)dr.
Let us pose
Then,
_-?(/)
Sy = SiJicosv/f + BS^i;sinv/.
(/) cin
(7.28)
where
1-
V2-
" _2
Bii)
-2«,^.
m
l^p—
L ^^
/'jr-
^iiy»
V2 -
2 _ v2
^~ .2
-f4ny 2
L ^/.
/';'
' 71
"Bii>+2n,-^B(i^
¥P-~
^^J
sr pj
^dyn
r ^2 1
2 _ v2
1--2
+ 4/2,
L ^^j pj
8<i
c
1
(7-29)
(7.30)
are the coefficients of quasi-static "blade deformation.
Equations (7*29) permit determining the dynamic coefficients of blade de-
formation if the quasi-static coefficients of deformation ohtained for the aero-
dynamic loads Tv and Tv are knovm.
6. Aerodynamic load on a Eigid Blade
In flight, a helicopter blade is acted ipon "by variable loads with frequen-
cies that are multiples of the rotor rotations. In this case, as already men-
tioned, the greatest variable stresses in the blade are caiised by the first six
to eight harmonics of the aerodynamic load relative to the rotor rotations.
Higher haiTnonics xisually are so small as to cause no noticeable stresses in the
blade, even in resonance.
A calculation of the variable aerodynamic loads on a blade encounters cer-
tain difficulties. These have to do primarily with the necessity of determining
the variable induced velocity field, consideration of nonllnearity in the de-
pendence of the aerodynamic coefficients on the angle of attack of the profile,
the Mach number (M), and coipling of the loads with the torsional vibrations of
the blade. The consideration of these characteristics is discussed in the re-
spective Sections. Here, we will construct formulas for determining variable
85
aerodynamic loads acting on a "blade rigid in flexure and torsion, imder the fol-
lowing asstoiptions •
a) We assume that the inlet angle to the t)lade profile $ (Fig. 1,26) is 781
small so that we can set approximately:
(7-31)
where
I = inlet angle;
Ux and Uy - mutually perpendicular conponents of the relative flow velocity
lying in a plane nonnal to the blade axis (see Fig#1.26). Here,
the velocity U^ is parallel to the plane of rotation of the
rotor.
;f Zapping
plane
Fig. 1.26 Diagram of Flow Past a Blade
Rigid in Flexure and Torsion.
Assimiing also that cos $ = 1,
we will consider that the unknown
load T acting in the flapping plane
does not differ from the load T^y ,
perpendicular to the inlet to the
blade profile (Fig. 1.26).
t)) We assume that the magni-
tude of relative velocity of the
flow (U) past the profile differs
little from the quantity U^ :
U^U,
(7-32)
c) We assume that, in deter-
mining the loads in the flapping
plane, the profile drag can be
neglected and that we can set Cx =
- 0.
We stipulate that the profile lift coefficient Cy depends linearly on the
profile angle of attack a :
Cy^C-y'O..
(7-33)
d) We assume that the induced velocity of the flow v passing through the
rotor is constant over the entire area swept "by the rotor:
t? = const
(7-34)
With these assunptions, only the constant portions of the first two har-
monics of aerodynamic forces are of substantial magnitude, and then only at me-
dium and high flying speeds of the helicopter. The high harmonics are small and
their calculation under the above assunptions is of no interest.
86
Using these assijnptions, the linear aerodynamic load on the blade can be
determined by the formula
2 y^ ^' (7.35)
We will assiJine furthermore that the profile angle of attack is
«=9,+^. (7.36)
where cpj. is the angle of blade profile setting in a section at distance r from
the axis of rotation.
Then, eq.(7«35) can be transformed into the form /82
T=\c^Jib [^rUl^UJJ,]. (7 .37)
For an ideal flexurally rigid blade, suspended at the hub by a flapping
hinge, the velocities entering eq.(7*37) c>an be determined by means of the fol-
lowing formulas:
Here,
Po = flapping angle of the blade relative to the flapping hinge;
3Po
Po = = time derivative of the angle Pq ;
dt
\o = relative velocity of the flow through the rotor;
where
Qfj.0^ = rotor angle of attack at the shaft axes;
Vq = induced velocity of the flow referred to ouR, which is constant
over the rotor disk#
The blade angle can be written as
qpr = 8 0+ Acp— 9 1 sin 115—02 cos i}3— "/Po, ( 7 -39 )
where
9q = blade angle at the relative radius r = O.? or at any other
radius adopted for reckoning 0o at Pq = 0;
87
Ill ill ill
Acp = geometric twist of the tflade;
9i and 83 = angles of cyclic pitch control of the "blade prescri'bed tiy the
swashplate •
If we represent the flapping motion of the blade in the form of a series
and retain there only the first two harmonic conponents, since the higher har-
monics are small at the adopted assimptions, then eq.(7-37) can "be transformed
into
T^=-T<^yQb,.-,'-'^'
2 "
{P^ COS rt'^-^-Pn sin n^)
(7.41)
where
(7.42)
^2 = ^[--^t^^^o4-K6t"272a2 + x(72+Yi^')^2
In performing these transformations we iised a substitution that permits /83
changing over to the so-called equivalent rotor.
An equivalent rotor is a rotor whose shaft is imagined as turned relative
to a real rotor through an angle such that the same angles of attack of the blade
sections are achieved without cyclic pitch control. All formulas written out for
an equivalent rotor can "be used without change for a real rotor without an auto-
matic pitch control. An equivalent rotor usually is also given the properties
of a rotor without a flapping conpensator. In this case, the formulas are
equivalent only to an accuracy to within the first harmonic of flapping.
Transformation of the formulas for aerodynamic loads, as applied to the
equivalent rotor, was performed under application of the following substitutions:
(7.43)
88
where
9r6*i - real iDlade angle with consideration of the effect of a
flapping conpensator at the radius adopted for reading
this angle;
ai, bi" and X© = flapping coefficients and relative velocity of flow through
the blades for an equivalent rotor*
Here, we will not discuss higher
harmonics of the aerodjmamic load*
Figure 1.27 shows the constant
portion and cosinusoidal and sinxis- l&k
oidal conponents of the first two har-
monics of the aerodynamic load, for a
typical helicopter "blade derived from
ecp#(7»42) for horizontal flight of a
helicopter at |j, = 0.28.
In Figs. 1.28 and 1.29 these loads
are summed; the diagram also gives the
total relative aerodynamic load P
acting on a "blade in the longitudinal
plane of the rotor at ilf = 0*^ and '^ =
= 180° (Fig. 1.28) and in the lateral
plane at ^Ir = 90° and tj; = 270° (see
Fig.1.29).
-0,03
Fig. 1.27 Distril3ution of Harmonic
Con^onents of Aerodynamic load over
the Blade Radius, for iJ, = 0.28.
7- Determination of the Blade
Flapping Coefficients
To determine aerodynamic loads ty
means of eq.(7.42), it is necessary to
know the flapping coefficients of a flexurally rigid "blade.
The flapping coefficients can be determined from the differential equation
(7.1) if we represent the solution of the equation in the form
where y^°^ is the mode of blade vibrations with respect to the fundamental.
For a rigid blade, this vibration mode coincides with a straight line
If the distance from the axis of rotation to the horizontal or flapping hinge /85
is equal to zero (l^.h = O), then
y^^^=-r.
(7.45)
89
Fig ♦128 Relative Aerodynamic Load Acting on a Blade
in the Longitudinal Plane of the Rotor.
which is valid both for rigid and elastic iDlades (see Sect .4)*
Setting i^^i^ =0, let us substitute eq.(7^45) into the differential equa-
tions (7.1) and apply Galerkin^s method to it. This operation leads to a dif-
ferential equation of flapping vibrations of the blade
'CPo+^'h)=-]Tdn
(7.46)
where I is the moment of inertia of the blade 'relative to the flapping hinge.
Equation (7.46) can also be derived by equating to zero the moment of all
forces relative to the flapping hinge.
Substituting eqs.(7-40) and (7«4l) into eq.(7»46) and equating the coef-
ficients of like harmonic azimuthal functions, we obtain a system of equations
from which we can determine all flapping coefficients. This system is written
out as a table (see Table 1.6).
Each equation of the derived system represents the sum of the products of /86
certain coefficients, entered in the squares of Table 1.6, while the unknown
flapping coefficients of the blade simultaneously entering several equations are
set apart vertically in a separate row above the table. The known coefficients
of each equation occipy one row in Table 1.6. On the right-hand side of the
90
table, a special column contains the coefficients $ making i:p the right-hand
side of the equations. The enpty squares of the ta^ble correspond to coefficients
equal to zero.
Fig. 1.29 Relative Aerodynamic Load Acting on a Blade
in the Lateral Plane of the Rotor.
Table 1.6
°-i
^f
if
'^2
h
'//
l/i^HC
-1,..
/iB
-A-i/c'C
1„
-flKB
A-lj^^C
y.a
i^s
-/IB
^-V.{A^yc) -ZA
4 !
l,^c
-fi8
ZA
f^-i'H/'c;
A.^^M^jV^^'
91
Illlllllllli
The follovdng notations were used in compiling the table:
6
1
1 _
1 _,
^9 = p?3^^arr.
J "^
1
{l^kl)
(7-48)
The mass characteristic of a rigid blade y is determined by the e^qjression
(7^49)
Y =
o^Qb^jR^
2/
On solving this system of equations, it is foiind that^ the coefficient ag
and bs are appreciably smaller than the coefficients ap, ai", and bf". Thus, they
can be neglected in determining the coefficients ao, bi, and a|\ This assunp- l&J,
tion leads to sirrple formulas for determining the flapping coefficients of the
blade
where
^o==Y
1
/*^+>^^ + ^tl2C<P
Of. — \ " •
*:==■
fxBao
A^ — ^^C
<Zn:
18 + 8>12y;
S^ (^4^56t+^a:)-(x2(2^Cao+^'j];
^^=i8il^ K-^^^^'^+f -^O+^l'^^-vS]
(7.50)
92
|-xv(^+^(.2c)'
8. Simplified Calculation of Elastic Blade Vibrations
Based on the sinplifying assunptions adopted in this Section, we can con-
struct the calculation of elastic vitirations and bending stresses in a blade for
horizontal flight regimes of the helicopter. Such a calculation, of coijrse, can-
not give positive results when applied to low flying speeds where a major role
is played by variable stresses having to do with the nonuniform induced velocity
field; the same is true for high speeds where it is inpossible to disregard the
nonlinearity of the dependence of aerodynamic coefficients on the angle of at-
tack and phenomena associated with flow conpressibility.
In conformity with the above formulas, the calculation is conveniently per-
formed along the axes of an equivalent rotor.
The calculation of elastic blade vibrations is carried out in the following
sequence:
1. First, determine the parameters of the flight regime at which the cal-
culation of stresses is to be carried out. These are the following parameters:
a) rotor angle of attack a^q ;
b) angular velocity of rotation of the rotor uo;
c) altitude and flying speed represented in the calculation by the
coefficients p and p..
2. Calculate the relative velocity of flow through the rotor from the
formula
'^^ 4K^M^' ' ^7.51)
where C^ is the thrust coefficient of the rotor.
3. Next, calculate the blade angle at the control section relative to which
the geometric twist of the blade is prescribed.
V\fi.thout consideration of forces related with the second harmonic of flap- /B8
ping, this angle can be determined by the formula
Here,
93
6
(7.53)
t = coefficient of thrust;
a = solidity ratio of the rotor.
4* By means of eqs.(7»50), determine the flapping coefficients of the blade,
and by means of eqs«(7.41) 3,nd (7-42) the external loads on the blade.
5. To determine the bending stresses, calculate the natural vibration modes
and frequencies of the blade.
6. If such a calculation is performed, eqs.(7-30) mil yield the quasi-
static coefficients of deformation with respect to different harmonics of blade
vibration from the constant conponent of the first and second harmonics of the
aerodynamic load.
Substituting eq.(7«41) into eq.(7.30), we obtain the values of the quasi-
static coefficients of deformation with respect to the j-th harmonic blade vibra-
tion
=^-y>Y'^,-VA)+\^'c)
'=vr'
(7.54)
Here, the subscripts of the coefficients of quasi-static blade deformation
correspond to the order of the harmonic of the aerodynamic forces. 'The index j
denotes coefficients pertaining to the j-th overtone of blade vibrations; Yj is
the mass characteristic of the blade in deformations vdth respect to the j-th
overtone :
Y/ =
l/2c;QbojR2
(7.55)
94
The following notations are adopted for the integrals entering eqs*(7*54) : /^9
(7-56)
where j^^ is the mode of blade vibrations with respect to the j-th overtone
normed such that y^^^ =1 for r = 1,
7* Then, write the blade deformations in the following form:
Z/ = [^o — ^iCOs6 — ^j sin — ^2^03 20 — 0^2 sin 2'^] £/(!) +
+ K — ^icoso — /isin4> — ^2C0s2;^ — /2Sin20]^(2)_i_
(7.57)
Here, in determining the blade deformation, the mode of the fundamental
which, in the case of ro = 0, coincides with a straight line, is replaced by the
first three harmonics of natural vibration of the blade j^^^ , y^^^ , and j^^^
noimed such that y^^^ = R at r = R. Then, the coefficients of blade deformation
entering eq.(7*57) can be determined in terms of the quasi-static coefficients
of deformation in accordance with eq.(7»29)-
As a typical exanple, let us write out the formulas for determining the co-
efficients of deformation with respect to the first harmonic:
^0 ^ ^0 J
^i = -
6>2
^— (0 -f.
Pi
1--
p\
* .-2 "'^
(7.58)
95
d,=.
<ii2
'-jrr*"+'"'^-''"
[-t]
^9=-
^.
40)2
^ — 2
ifi)
1-^
4to2T 4ai2
40121- 20) -/TV
5a)_2nL- — ^^'^
1-
4o)2 -
"A
.2 4«2
790
(7.58)
If the dynamic coefficients of deformation are known, it is easy to deter-
mine any conponents of the stresses set ip in the blade. This will "be discussed
in more detail in Subsection 1? of Section 8 and in Subsection 8 of Section 9*
In the sinplified method of calculation presented here, a large number of
additional assuirptions of a conputational nature applicable to almost all stages
of the calculation are used in place of the initial assi:inptions pertaining to
the physical properties of a blade model adopted in deriving eq.(7»l) and in cal-
culating the right-hand side of this equation, which reduces to eq#(7-35)« All
these sinplifications, although they make the method of calculation quite suit-
able for manual corrputation, introduce numerous indeterminacies that are poorly
amenable to a quantitative evaluation. Despite this shortcoming, the described
sinplified method of calculation has one important advantage, namely its clear
presentation. In principle, all calculation results obtained by other more im-
proved methods are evaluated and analyzed on the basis of dependences presented
here in a sinplified form.
However, even with the use of all these assimptions, pencil-and -paper comr-
puting by this method takes one month of work for one calculator. The current
flow of blade designing cannot be maintained when one calculation takes that
long. Therefore, the calculation of elastic vibrations of a blade used for se-
lecting the blade design parameters can be performed only on high-speed elec-
tronic conputers. Naturally, there is no need then to use assuirptions that fa-
cilitate the coiiputational process •
Consequently, in Section 8 we will present a method of calculation based on
the same initial assunptions, provided we neglect all variable induced veloci-
ties; the method uses no assuirptions of a conputational nature.
96
Section 8. Calculation of B endii^ Stresses in a Blade at Low and Moderate
Flying Speeds
1. Charact eristics Dist ing uishing Flight Regimes at Low
and Moderate Speeds
Low and moderate speeds of a helicopter are regarded here as regimes suf-
ficiently remote from flow separation in which, furthermore, phenomena associated
with flow coirpressi"bility can 'be neglected. On this T^asis, in calculating aero-
dynamic loads it is assumed approximately that
'y-^l'^^' (8.1)
This assunption greatly simplifies the calculations necessary for construct-
ing design formulas.
On the other hand, low-speed modes can "be regarded as regimes especially /91
detrimental to fatigue strength and often conducive to the generation of maximum
bending stresses in the "blade.
These considerations justify the use of a method of calculation suitable
only for low and moderate flying speeds but not for high speeds nor for regimes
in which phenomena associated with the nonHnear character of the dependence Cy =
= f(cy) and with flow coirpressibility become determining factors.
It should be noted that the assunption (8.1) does not always hold for low-
speed modes. In cases in which the rotor blade accounts for an extremely large
load, the calculation should be performed with consideration of the nonlinear de-
pendence of the aerodynamic coefficients on the angle of attack of the profile.
The method of such a calculation will be discussed in Section 9*
The bl^de overloading can be estimated from the value of the thrust coef-
ficient of the rotor t. Calculations show that the assunption (8.1) can be used
for low-speed modes without introduction of substantial errors into the results
at t < 0.18.
In regimes with vertical overloads such as, for example, the braking regime
of a helicopter before landing, an infringement of this inequality might occur
in rotors which show such overloads in steady flight. AH this must be taken in^
to account in selecting the calculation method.
2. Method of Calculating Stresses
This Section presents the conventional method of calculating variable
stresses, based on Galerkin^s method with expansion of the deformation coeffi-
cient in a Foiorier series in harmonics.
Because of the possibility of using this method for calculating low-speed
modes, the harmonic conponents of the induced field are introduced into all cal-
culation formulas, and the problem of blade deformation is solved simultaneously
97
with the prot)lein of deterininLng the induced velocities •
However, such an approach is not a "must" for the method proposed here. In
calculating stresses at moderate flying speeds when the variable induced veloci-
ties do not cause excessive refinements in the results, it can "be disregarded.
In this case the method of calculation is greatly sinpHfied.
If the assunption (8.1) is used, the aerodynamic load will be a linear
function of the displacements of the blade element, and the problem of calcu-
lating the bending deformations will reduce to solving the linear differential
equation (1.9). To solve this equation we use the B.G.Galerkin method. The
blade deformations are represented as a series with respect to eigenf unctions,
while the time coefficients of this series are expanded in a Fourier series. The
use of Galerkin^s method transfonns the differential equation of blade vibrations
into a system of algebraic equations relative to the unknown coefficients of the
Fourier series, and the determination of the blade bending deformations reduces
to a calculation of these unknown coefficients. Such a method of calculation
will be presented here.
3 . Assumptions in Determining Induced Velocities
When calculating the bending stresses at low flying speeds when their value
is determined mainly by the degree of nonuniformity of the induced velocity /92
field, the assunptions on whose basis this field is determined become of great
inportance.
In the first volume (Chapt.II, Sect.5), it was mentioned that induced ve-
locities can be represented as the sum of the extrinsic and intrinsic induced
velocities. This subdivision is somewhat arbitrary but proves useful since it
permits an evaluation of the effect of individual induced velocity conponents by
analogy with the evaluation conventional for the wing of a regular aircraft; this
justifies the adoption of certain assunptions inportant for further presentation.
The flow past a helicopter blade with a nonuniform induced velocity field
is analogous to the flow past the wing of a regular aircraft in flights in turbu-
lent air, when the wing constantly encounters airflows of differing velocity and
direction. During rotation of a rotor, the blade also encounters in its path a
nonuniform velocity field, except that this field is not caused by atmospheric
turbulence but by the induced action of the entire vortex system of the rotor.
This field, by analogy with a wing, is usually called the extrinsic induced ve-
locity field, unlike the velocity field induced in the blade region by the vor-
tices shed by the blade due to a change in circulation with respect to time and
blade radius. That these vortices create appreciable induced velocities at the
blade is due exclusively to the fact that they are at a very short distance from
it. Upon removing the vortices a distance of 20 - 30° from the rotor azimuth,
their influence on the aerodynamic load on the blade will decrease.
Just as in calculating a wing, the "steady-flow hypothesis" can be used in
determining the aerodynamic loads on a blade. According to this hypothesis it
is assumed that, in a nonsteady flow past a profile, the loads acting on the pro-
file behave as though the flow pattern produced at a given instant of time would
98
remain unchanged for an arbitrary length of time. In conformity with this hy-
pothesis, in calculating the aerodynamic loads on a wing allowance is made only
for the change in angle of attack produced by the extrinsic velocity field, while
the effect of the intrinsic induced velocities is disregarded •
We will use an analogous approach for the case of a blade. In determining
the aerodynamic loads, we will take into account only the extrinsic induced ve-
locity field.
In the calculation of this field, certain additional assumptions relative
to the characteristics of the vortex system in the low-speed mode can be used.
Figure 1.30 gives a planview of a system of free vortices shed by the blade
tips of a five-blade rotor in a flight
regime with a speed corresponding to
fji == 0.05 • At this speed, the variable
stresses in the rotor blades reach a
maximum.
Direction of flight
Fig .1.30 View of a Vortex System
Shed by the Blade Tip in the |j, =
= 0.05 Regime.
The picture conveyed by this
sketch is incomplete, since only free
vortices shed from the blade tips are
shown while the vortices shed from all
other blade radii are omitted. The
radial (transverse) vortices are also
left off. However, even this pattern
already gives an idea on the close
spacing of vortices in low-speed
regimes. Due to this characteristic
of the vortex system, the induced
actions of individual vortices will
merge and appear as the total nonuni-
formity of the entire velocity field.
No sharp induced velocity peaks, char-
acteristic for the vortex system, with
widely spaced vortices occur. There-
fore, at low flying speeds and especially for rotors with a large number of
blades, the induced velocities can be determined from the theory covering the
configuration of a rotor with an infinite number of blades.
With an increase in flying speed, the free vortex system starts extending /93
and shows wider spacing. The vortex system also changes in the same sense on a
decrease in the number of rotor blades. This reduces the accuracy of calculation
for a configuration with an infinite number of blades.
On changing from a given rotor to a configuration with an infinite number
of blades, the local effect due to the vortices immediately adjacent to the blade
is reduced so greatly that, in first approximation, it can be assimied that this
design does not allow for the influence of adjacent vortices so that the velocity
field determined on its basis will practically coincide with the extrinsic in-
duced velocity field.
99
nil
mil I II iiiiiiiiiiHiiiiinii I mill III
iiiniiiii
The considerations presented atove lead to the conclusion that, for calcu-
lating elastic Tolade vibrations at low flying speeds, the vortex theory iDased on
a scheme with an infinite number of blades can be used.
At low flying speeds, one usually measures variable stresses of which a
major portion is made up of high harmonics of the rotor ipm, generally located
between the fourth and sixth harmonic. Therefore, still another imp ortant re-
quirement must be inposed on the method of determining induced velocities. Such
a method should determine the induced velocity field with an accuracy of at
least to the sixth harmonic, which is possible only if the circulation values
are determined with an accuracy to the same hannonic. Consequently, all methods
not satisfying this requirement are worthless and cannot be used for calculating
elastic vibrations •
As stated above, we will present a method of calculating stresses in which
all variables are expanded in Fourier series in harmonics . Therefore, it is
convenient to use the method of determining the induced velocity field, in which
these velocities are determined also in the form of an e^q^ansion in harroonics .
These stipulations are best met by the theory developed by V.E.Baskin
[(Ref.3); see also Sect. 5, Chapt.II of Vol.1]. Therefore, this theory will be
used here for our stress analysis.
4* Mathematical Formulas for Induced Velocity field Determination
Let us examine the system of formulas proposed by V.E.Baskin for calculating
the induced velocity field in the plane of rotation of p rotor.
We will represent the field of these velocities as the sum of its harmonic
conponents. In so doing, both the total flow velocity and the harmonic comr- /9U
ponents of this velocity are related to the tip speed of the rotor blades cdR:
X = !i./a/7a^^Ao-f2(>^^cos/^;j-^X^sin/i'^). (g,2)
Here,
X = total velocity of the flow passing through the rotor, relative
_ to cjuR;
^0 " constant induced velocity conponent, also relative to cuR;
\^ and Xn = harmonic induced velocity coriponents;
I = azimuthal blade angle reckoned from an axis coinciding in di-
rection with the tail boom of a single-rotor helicopter;
Vcos arct
where
V = flying speed of the helicopter;
Q^rot - rotor angle of attack at the shaft axes.
100
The linear aerodynandc load acting on the blade is represented in the form
r=-i clQbo,y>»WP,
(8.3)
where
Cy = angle of slope of the dependence Cy = f(cy), which here is taken
to "be linear in the form of eq.(8.l);
p = air density;
tiQ.? = value of l^lade chord at the relative radius r = 0.7#
Henceforth, the value of P entering this e:xpression will "be designated as
"relative aerodynamic load" •
¥e represent the value of P in the form
(a.4)
The harmonic velocity conponents \„ are represented as the sum of the so-
called partial induced velocities, each of which is induced only by one harmonic
of the aerodyTLamic load
m
(8-5)
In these expressions, the sum total induced velocity conponents have one
subscript n, while the partial conponents have two subscripts n and m.
The values of the partial harmonic induced velocity conponents are deter-
mined by the following e:>q)ressions :
at /i = 0:
3^'o™= -^^^^0 (- ir-r^y (PJ;
at n ^0:
(8.6)
If the power to which t is raised is negative (n-m<0), we miist set /95
= (-t)"~° in ecp.(8.6).
The coefficients entering ec|s.(S.6) have the following values:
101
Ao =
T=z-
0=1
(8.7)
where
a = solidity ratio of the rotor;
Zt == nimiber of blades of rotor.
The value of the disk flow ratio averaged over the blade radius X^av ^s de-
termined from the formula
>^ba.=l*f^''«/^ri-2fXoFcfr.
(8.8)
To determine the functions J(Pn) and J(Pn ) entering eqs.(8.6), the follow-
ing formulas are obtained from V.E^Baskin's theory:
JiPJ"
HPm) =
^]jn{zr)z J
\P„{Q)J„iZQ)dQ
\P„(Q)J„{ZQ)dQ
]dz;
]dz.
(8.9)
where _ _
Jn(zr) and JaCzr) = Bessel functions of the first kind of order n and m,
respectively;
z = integration parameter •
Here, to specify the parameter over which integration is carried out, a new
notation is introduced for the relative blade radius p"# This notation will be
used only in calculating the integrals (8.9)*
5. Transformations of Mathematical Formulas in Particular Cases
Equations (8.6) are greatly sijiplified in particular cases. Thus, in the
case of n = m = 0, we have
^00 icloA,^.
(8.10)
For further calculations, the result obtsiined for the case of n = m is es-
pecially inportant. It will ise found that the coinciding harmonics of the aero-
102
dynainic load and induced velocity are ;iniquely related ty the expressions: /96
(8.11)
where
r 1
(8.12)
ClaAo[l~(-irT^^l^
This formulation makes it expedient to separate the induced velocity couk
ponents into two types: principal induced velocity conponents due to the same
harmonic of the aerodynamic load as the harmonic of the induced velocity, and
secondary conponents due to all other hannonics of the aerodynamic load.
This separation permits writing eqs.(8.5) in the form
(8.13)
where the principal induced velocity conponents are determined by eq.(8.11),
whereas the sxmi of all secondary induced ve_locity ^onponents is introduced into
the equation ty means of the new notations XI and X^:
m^n —I
m—^.
m = m^n + l
m™/T— 1 m=z,
^!:= 2 ^
ft-.
m=0 OT«n-fl
(B.2U)
Here, z^ is the niomlber of harmonic induced velocity conponents taken into
account in the calculation.
At n = 0, the first memt)ers of these expressions should be equated to zero,
and at n = Zjj the same should be done with the second members. In constructing
the equations for stress analysis, the induced velocities will be represented in
the form of eq.(8.13).
6. N-umerical Determination of the Integrals J(Pb) and J(P^)
At m ^ n, a calculation of the integrals (8.9) encounters certain diffi-
culties. To determine the values of these integrals, V#E.Baskin proposed a
103
method in which the aerodynamic load conponents are approximated by trigono-
metric polynomials- For this, it is necessary to determine the values of P„ at
prescribed "blade radii not coinciding with those used in the overall calcula-
tion* This is not too convenient for the method proposed here. Therefore, we
will use another method more suitable for the given case, in which calculation
of the integrals J(Pm ) and J(P,) is carried out approximately in the same form /97
in which the integrals are conputed when calculating the blade stresses. To
this end, the blade is divided into individual segments witMn whose limits the
aerodynamic load is represented in a form sioitable for integration. Here, it is
logical to divide the blade into the same segments in all cases, both when cal-
culating the stresses and when calculating the integrals (8.9) • We will repre-
sent the load P„(p) such that, at each segment of integration, this load will
vary in accordance with the law
r:\ ^miQk) "m+l
PmiQ)--^
ex
m-M
(8.15)
Here,
P =
Py =
current values of relative blade radius; after integration and sub-
stitution of the limits, the value of p" will no longer be contained
in formulas without an index;
same value of relative radixis but with the subscript k, which means
that the radius in question coincides with the radius at which the
relative aerodynamic load 'P^(p^) is calculated.
OJ
OJ
Pk^i
1
1
p^
pK-1 -
'M
^
/
n 1
1 1
T
H
/
/
f\
1 1
^L.i.l
i ii>
\
V
/
, I ' .
' _[
J._.
1 i
1
N
\
1 ,
. f
i\
/\
1 ,
w
/
1 1
1 1
/ (
/ '
/ 1 1
J 1
1 1
1 1
1 *
1 1
[-} r
[ i. 1
i 1 * f ' ? » V
1 ! 1 :MK
1 r 1 t I 1 1 l\
0.2
OA
0.6
as
p
Pig. 1.31 Shape of the Relative Aerodynamic Load Adopted
for the Calculation of Induced Velocities.
Henceforth, as already stated, let us differentiate the relative radii pj^ ,
at which the value of the aerodynamic load is taken, from the relative radii r^
104
at which the induced velocity is calculated. This prevents possible confusion.
Let us assume that the relative aerodynamic load varies in accordance -with
the law (S.15) over the length of each segment bounded "by the relative radii
In Fig. 1.31^ the solid stepped line gives the shape of the distribution of
the relative aerodynamic load over the "blade length, represented for calculating
the induced velocities from eq.(8.l5) in the case m = 0. Such a form of repre-
sentation of the aerodynamic load naturally may introduce certain errors into
the values of the induced velocities. However, calculations performed to es-
timate the magnitude of this error demonstrated that the error is small and is /98
unable to cause substantial changes in the calculation results.
On substituting the value of the relative aerodynamic load expressed in the
form of eq.(8.15) into the e^q^ression of the integrand of eq.(S.9), then the
interior integral on the right-hand side of this equation can be represented as
some siom of definite integrals:
The definite integrals entering this e^q^ression can be calculated anal3rti-
cally [see (Ref .11)]. Substituting the integration limits into the obtained ex-
pressions, we can write:
1
where
* 2
_gmiO*)__ ^m("Qt+l) "
Q7*' Q^l J
Substituting the resultant value of the interior integral into eqs.(8.9),
we obtain
(8.18)
Or, if we write this in a singjler form,
105
lllllll
mil II ■■■iiiiiiiiiii
where
The integral (8.20) is a discontinuous integral known as the Welser-Schaf-
heitlin integral (Ref.ll). ^Its analytic expression, as a function of the rela-
tion between r^ and -^(pi^ + pk+i ), has the following form:
If 7 (Qft + Oft+iX'^i* ^^^'^
-^An)=[jAzP)J.4zl^)dz J ' J '^ fn~J ^ (8.21)
I* 2-(0ft+0*+i)>O. then ^
0'
1'(^±|^) (8.22)
L 2 2 VOfc+CA+i/J
Here,
r = gamma function with different arguments;
F = hypergeometric function of the argiment a, P, y* z*
These arguments, as indicated "by eqs.(8.2l) and_(8.22), may have different
values depending ipon the relation between v^ and ^pj^ + pj^^^ ). For exanple, in
eq.(8.2l), we have
q^ m4-2 + n ,
2
^ 2 '
Y = m-{-2;
106
When performing the calculation on a digital conputer, these functions are
easy to program. Therefore, their calculation presents no difficulties.
?• Assumptions Adopted in A erodynamic F orce Determinations
In determining the aerodynamic loads, the assumption (8.1) is siJpplemented
"by the same assunptions used in detemdning the rigid "blade loadings (Sect. 7,
Su"bsect.6), with the exception of the assxanption (7«34)»
1. Let us assume that the inflow angle to the "blade profile $ is small and
that we thus can asstjuie approximately:
(S.23)
where
U^ and Uv
= inflow angle;
= mutually perpendicular conponents of the relative flow velocity
in a plane normal to the elastic iDlade axis (Fig.l.32); here,
the velocity Uj^ is parallel to the plane of rotation of the
rotor.
Plane of
rotation
Axis of rotor
shaft
Plane
p € rp endi cut ar
to elastic
blade axis
2. Let us assume that the
magnitude of the relative ve-
locity of circulation flow U
around the profile differs lit-
tle from the magnitude of U^-
Therefore, we can assume that
us u,.
3» Let us assume that, in
determining the loads in the
flapping plane (plane going
through the axis of rotation of
the rotor), the profile drag
can be neglected and it can he
assumed that c* = 0.
Fig. 1.32 Diagram of Flow Past a Blade, Used
in Stress Analysis at the Low-Speed Mode.
4» Stipulating that cos $ =
= 1, let us assume that the /lOO
load in the flapping plane does not differ from the load peipendicular to the
inflow to the blade profile (see Fig.l.32).
8. Mathematical Formulas
When using the assumptions given in Subsection 7* the value of the relative
aerodynamic load P entering eq.(S.3) can be determined from the formula:
P = ^b(Jx + U^(^y].
{&»2k)
107
Ilillillll
where
"Bp = value of the TDlade chord at the radiias in question, relative
_ _ to the chord at a radius r = 0.7;
Ux and Uy = same relative flow velocity conponents as in eq.(8»23) "but
relative to the tip speed of the rotor t>lades ouR:
Z7^=:r+ixsint|>; 1
Here,
X = relative velocity of the flow through the rotor; this velocity
is determined from eq.(8»2);
y = displacements of the elastic "blade axis in a plane perpendicular
to the plane of rotation from which these displacements are cal-
culated;
P = y' = angle of slope of the elastic "blade a3d-s.
Here the prime denotes differentiation with respect to ,the blade radius and
the dot, with respect to time.
The "blade setting angle can be written in the form
9 = 6o + Ao-6iSirn;>-^62COS'{^-XoPo' (8'26)
Here, __
9o = "blade setting angle at a relative radius r = 0.7 or at some
other radius adopted for calculating 0o, when the angle of rota-
tion of the "blade in the flapping hinge Po is equal to zero;
Acp = geometric blade twist;
01 and 02= cyclic pitch control angles prescribed by the automatic pitch
control;
H = flapping conpensator;
Po = angle of rotation of the blade in the flapping hinge.
Let us represent blade deformations in the form /lOl
y=I^W^ (8.27)
where
6 J = coefficients of blade deformation corresponding to the j-th
harmonic of its natural vibrations; these coefficients are func-
tions of time and therefore are also called time factors;
y^j^ = natural blade vibration modes in vacuijm normed such that
,.;» - E.
Let us expand the time factors 6j in a Fourier series in harmonics. Then,
the blade deformations can be represented as
108
^ = [^0 — 2(«nCOS/^4> + ^'„sin/^6)jz/to)-|-
+po— S(<^rtCOs/i'}.+flf^sin/t»l»)lt^O)+
(8.28)
+ [g'o-2(^«cos/i']* + A„sinrt^)1-£/(3) + ...
This form of notation of the solution is a continuation of the conventional
form of notation for the flapping motion of a blade (7»40).
After differentiating eq.(8.28) "with respect to radius and time and sub-
stituting y, together with its derivatives and eqs#(8.2) and_(8.26)^ into
eq.(8.25), and then substituting the resultant formulas for U^ and Uy into
eq.(8«24), we finally obtain the e^^ression from which all harmonic conponents
of the relative aerodynamic load P can be determined.
These conponents can be represented in the form
(8.29)
Here, f^ and f^^ are certain functions determining the value of this com-
ponent of the aerodynamic load, which does not depend on the magnitude of the
induced velocities-
If now the induced velocities X^ and \^ are represented in the form of the
sum of the main and secondary conponents and if the main conponents are expressed
in terms of P^ and Pj^ with respect to e(^.(8.1l), then the values of P^ and P^
will appear both on the left- and right-hand sides of eqs.(8.29)-
After determining from these equations the values of P^^ and Pj^, we ob- /102
tain the following expressions:
Pn = B. [f + Tkl - -L J. (X„_, -\^,) j ;
(8.30)
where
109
B — ^'
l4--^c;a>lo[l+(-l)V'']^
^ =^ br
l + Y^^^0ll-(-irt^"]^r
(8.31)
Below, the values of B^^ and Bjj will "be denoted as equivalent chords of the
"blade since, in the calculation, they play the same role as the actual chords
and, in eqs#(8.30), appear at the same place at which the values of "bj. are lo-
cated in eqs.(8.29)*
Thus, the harmonic coirponents of aerodynamic loads, with consideration of
variatfle induced velocities^ should t>e determined "by substituting only secondary
induced velocity coirponents into the formulas and replacing the real chords by
the equivalent blade chords. The values of the equivalent lolade chords may dif-
fer depending on the fHght regime and on the order of the harmonic of aero-
dynamic load "being determined. However, they always prove to be smaller than
the real chords. Consequently, all harmonic aerodynamic load conponents are
smaller than the values they would have if the main induced velocity coirponents
were equal to zero and are also smaller as many times as the equivalent chords
are smaller than the real chords. The introduction of equivalent chords leads
to a decrease of all aerodynamic load coirponents, "both exciting and dairping the
blade vi"brations. Therefore, there will also bie a decrease in the values of the
relative coefficients of aerodynamic danping which deteimnes the vib)ration
airplitudes in resonance. This causes a decrease in the variable blade deforma-
tions far from resonance, whereas those in resonance remain approximately the
same as in calculations without consideration of this effect.
Expressions of the type of eq.(8.30), written for all harmonic coirponents
of the aerodyr^amic load, are found to be interrelated over the induced velocity
conponents. Hence, these constitute a certain corrplex system of equations rela-
tive to unknown loads, which can be solved only if the values of f^ and f ^^ are
known. These values, however, depend on the magnitude of the coefficients of
blade deformation. Therefore, to solve this system of equations it is necessary
to construct equations for determining the defoxTnation coefficients. This will
be carried out below.
If the values of f^^ and f^ entering eqs.(8.30) are described in detail, the
esq^ressions for the harmonic aerodynamic load conponents can be represented in
the form of Table 1*7*
The expression for each harmonic coiiponent of the loads Pj^ andP^ occipies
one row in the table and represents the sum of the products formed by the coef-
ficients entered in the squares of the table with the imknown factors simultane-
ously contained in several expressions and entered vertically in a special row /103
at the top of the table. These factors, as already mentioned above, are called
the coefficients of blade deformation. The right-hand side of the table contains
a n-umber of terms f§, f?, f^, and f^, not related with the unknown coefficients
of deformation.
110
To determine the values of P^^ and P^^ , it is necessary to multiply the sum
of the products of the terms of each row and the unknown coefficients of de-
formation, which sum is added to terms independent of the coefficients of defor-
mation, by the values of Bj^ and B^^. These values are entered on the left-hand
side of the ta*ble.
The number of terms entering the e:xpressions for P^ and P^^ depends on the
numl^er of harmonics and overtones of the natural vibrations being taken into ac-
coiint in the calculation. In Table 1«7, the eixpressions are given for the case
where only two overtones and four harmonics of the variable forces are taken in-
to account in the calculation.
In programs used for calculation on digital coiiputers, four overtones of
natioral vibrations and six to eight harmonics of variable forces can usually be
considered.
9 * Conversion to an Eqiaivalent Rotor
In order to demonstrate the possibility of converting to an equivalent
rotor, the following equality was used in coirpiling Table 1.?:
-(0) —
y =r ,
which is valid only when the distance between the axis of rotation and the flap-
ping or horizontal hinge i^^-^ is equal to zero.
If we now use the known formulas for the coefficients of flapping and angles
of attack of an equivalent rotor
a* = ai — xi^i + 6i;
then the expressions for P^^ and P^ can be somewhat sirrplified by substituting,
in the first row of the table, the values of a^^ and bi for ax and b^. This will
Q
cause the coefficients f^ to become equal to zero, and the values of the angles
01 and 2 will not enter into the equation. In other words, the well-known prin-
ciple that blade loading does not depend on the deflection of the automatic pitch
control at l^,^ = 0, is coirpletely observed in the expressions of Table 1.?-
However, the sinplifications obtained on converting to an equivalent rotor
are so insignificant as not to justify the assimption of l^^^j^ =0. Therefore,
we will investigate blade vibrations only in the axes of the shaft and will not
use the concept of an equivalent rotor.
10. Basic Assumptions Used inCalculation of Bending Stresses
In the calculation of bending stresses in a blade, we will use the assimp-
111
Po
So
h
B,
h
J 3
~llf
fir
fir it,
-11 fit,
[Zf
^2
-ft fa,
-2ri
J=0 "
t = J
"
2fi
ic,(fuiy
COS <J)
Pi
sin f
cos 2f
\
sin Zt/f
^2
cos Jf
h
sin 3f
cos ^f
^v
-if-'^o
-Ia^
sin H f
i^'
-h'^,
TABLE
J3
.*^__ _
H
■ **
• • •
-
1 ,
ff^^
-if^'^^o
U'
r
-fif
3ri
-V^
-{/^'^O
oir'^ht^')
"Ia^
ftrxg
o • «
-3fi
-firx^
-JFr
• • a
2ftr
2 fir
a • •
«,{^'*{f*')
jffZ
• • •
fif'^o
-ff^2
Xc(^^-^iF')
9 « •
• • •
• • •
• • *
...
.7
''z
J'1
"s
■*;
d,
^z
-yy^'^jiv/'
-ftfK,
h'M''
-/trjjO
x,(f^'^h')
./trx,
.f^i1)^^^2^1)
-fiy('i*j/tfj
{/^y('>*i;cf^<'^
-firtc,
2fy(f^
-lf,3^0>
/if it,
l[,yW,yf0
-Zryin
«,(r-^*|/.^;
-U'^1,
-l^^zji^n
-/zfK,
ifi'M'^
-^/'S
^j/f'>+j/fr^'
-l^Zj^V)
-,-A^«,
,
1
^1
, _ '^^ .
^',
• • •
__
'W
-f
-l/iy('>^j/ifj3('>
lifit^
'ItVH,
-\fiphl,iffi^u
.Xj(f2*^/i^)
3fy('>
• • •
'/ryO
• • •
')
• • •
~
fifH^
• • •
Terms, Independent of the Coefficient
s of Deformation
^:
^n
^:
• 4^
-/er0,
^rtsfXa
(r'*U')r^
^K'-ii'h
r'^i/c^)0.
fVl^ly.\
Cr-Vf/6';«,
feHg«.g
Zfiff^
rl';^^l^-lfil^
/ifOj
'■
-w%
f^-j^a,-h)
-/ifO^
fly-l^iw)
J/^'^Z
f^ll-l^(l^-l^)
i/^'o,
rl; + |^(VV
^"^rf/'cvv
rll-i/^Ch-h)
» • •
Hi
111 a
tions adopted in deriving the differential equation (I.9) of blade vibrations in
the thrust plane. We will represent the blade as an elastic beam extended by
centrifugal forces N* The parameters of this beam - its linear mass m and the
flexural rigidity EI - "will be considered as continuously distributed over the
blade length-
Furthermore, we are adopting the following assunnptions: /104
1. We will assume that the plane of minimum blade rigidity coincides with
the flapping plane, so that the blade will bend in the flapping plane only under
the effect of forces acting in this plane.
2» In determining loads in the flapping plane we will disregard torsional
blade deformations (see Sect.? of Chapt.IV in Vol.1, on consideration of tor-
sional deformation) .
3. We will assume the conventional type of rotor with hinged blades and dis-
regard the distance from the axis of rotation to the flapping hinge, i.e., we
will pose lh,h ?^ 0* W^e will also neglect the frictional forces in the blade
hinges.
11. Differential Equation of Blade Vibrations and its Solution
When using these assunptions, the calculation of bending stresses reduces
to solving a differential equation whose derivation has been given in Section 1
of this Chapter:
[EIy"r-[Ny'Y + my = T. (g^32)
With the blade attachment in question here, the boundary conditions can be
written as
[^V]o-0; [EIy'% = Q. f (^'33)
The value of the linear aerodynamic load entering the right-hand side of
eq.(8.32) is determined from eqs. (8.3), (B.4), and Table l.y.
After substituting, into this equation, the solution of the form of equa-
tion (8.28) and applying Galerkin^s method, we obtain a system of algebraic equa-
tions relative to the unknown deformation coefficients. This system of equations
is represented in the form of Table 1.8.
Each equation of the obtained system represents the sum of the products
formed by certain coefficients entered in the squares of the table with the un-
known coefficients of deformation simultaneoiosly contained in several equations
and entered vertically in a special row at the top of the table. The known coef-
ficients of each equation occtpy one row in the table. The right-hand side of
the table, in a special column, contains the coefficients ^^ and ^^ representing
112
lABLE 1.8
J-^0 I
J^i \
J^Z 1
J^J
n
i
^;
A|
^i
hloy
iL
C,
h
• ••
^Cq
C
it
ii.
d2
Sl
il
fi.
d^
-..
-*fl
^f f* '; fil^j h ^H ft *" -ggl
^ V i2, -2t ,2l -2: li. i?T ""
/=^
T
"T
U
L
n^
Q
S
T
U
L
K
Q
S 7 U L K Q S
'TUCK »,
£-
R
S
J
U
^?
R.
Q
S'
7
U
L
R
Q s r U L ft
S 7 U L ^^ .^
1
zw
R
5*1
~s.2
T
U
L
K
ZH
R
Q±
S*K
7
U
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QKSfK 7 U L K *(_
J
N
^
^^
S
T
u
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S
r
L
N R'U9-L S 7 U L ^ * A
R-UO'L s r U. L A
2
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7
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MNRQS7ULK M \ H \ r\ Q \ S \ T \ U \ L \ Ki \\ (Pz\
/=/ J^
7
ZH L M\ N
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!/,
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j- Iv' .V R 'ro\s''^ 7 U •" Sj_
7 1
\ K\ L
M N \ R ,Q
S ' T
...
K L M N
R Q
s
7 '-•
KLMHRQS7'"
Ui.M** '^ 1« .?J ^ r ... 0.
^f
S
1 s
L M N . R
Q^S
\ L M
n\r
s i---
\L M N R Q S •"
! ! U M /y A r^" /!-• Ji
S
_,! 1 !
/( ■ L ^ M ,
N
/?T1
'
\ f< ' L
m' fJ \ R > Q
Ik l m h r q *"
\ \ ' V.^J- ^ ^ 'A J^j"-' *Y
r-i ; i 1 1 1 P-
'•'
...|...
■
"•,-|--|'-'
-■«
' f"— ^ —
. . \ „ :
1 i ( 1 1 ; \$i.
the right-hand side of the equations.
Just as in Tables 1-7 and 1.9 there are enpty squares in Table 1.8. This
means that the coefficients of the equations for which these squares are in-
tended are equal to zero.
12- Determination of the Coefficients on the Left-Hand Side
of the Equations in Table 1.8
To determine the coefficients on the left-hand side of the equations, a
special operator must be devised for the coirputational program. This operator
should read out the values for all coefficients of any equation of the system.
To render this operator as siirple as possible, we will divide the entire table of
coefficients into a number of zones relative to the number of harmonics of natu-
ral vibrations used in the calculation. These zones are then used for combining,
into separate groiips, all coefficients that follow a similar path of formation
when using Galerkin^s method and can be roughly calculated by the same formulas.
The transformation of the differential equation (8.32) by means of Galer-
kin's method into a system of algebraic equations of Table 1,8 comprises the fol-
lowing operations:
1. Into the differential equation (8.32), we first substitute the solutio n/105
in the form of eq.(8.28) containing various natural vibration modes. If the
modes entering into the solution (8.28) are denoted by the subscript J, then all
terms of the equation obtained as result of this operation can be diArided into
several groups, each of which is characterized by a definite subscript J.
2. Next, all terms of the equations are multiplied in turn by the same natu-
ral vibration modes y^^^ . As a result of this operation, a system of equations
is fonned in which each equation differs from the others by the harmonic of the
natural vibration mode j^^^ by which all terms of the equations had been multi-
plied. Therefore, the resultant equations were numbered in accordance with the
values of the index I.
3. Integration over the blade length of all functions obtained as a result
of prior operations is the next step in Galerkin's method. As a result of this
operation, all terms of the equations which previously had been functions of the
blade radius become constants.
4. In the next step, each equation obtained in this manner can be divided
into numerous siupler equations if all coefficients of like values of cos n^lr and
sin ni|r are equated. As a result of this operation, each equation with the nu-
meral I will be transformed into an entire family of equations. The individual
equations entering this family are coordinated by the index i in Table 1.8.
Furthermore, each pair of equations pertaining to like harmonics is marked by the
index n, equal to the order of the corresponding harmonic.
An analysis of the resultant system of algebraic equations shows that all
like coefficients of equations are arranged diagonally in Table 1.8. This ar-
rangement is repeated in all zones corresponding to different indexes J and I.
113
It should be noted that there are exceptions to this rule, which becomes obvious
from a study of Table 1.8 •
Using the above indices i, n, J, and I, it is possible to construct general
formulas for all coefficients entering into the left-hand side of the equations
of Table 1*8# In constructing these formulas, we will use special functions
fi(a) and fsCa) which assume the following values, depending upon the parity and
magnitude of their argument:
1 at C even;
at a oddj
1 at a==0;
at a -7^0,
These formulas have the form
.hu-n
<d2
—^j[A,^\v--'CjY^-hm
^==-f i-Vi (»-}- 1) //y/-i*v/i ('•) Br,
Here. Xy = «oBW,
(8.34)
/106
(8.34)
where
Po"'^ = angle of rotation in the horizontal hinge during viferations vri-th
respect to the mode of the j-th harmonic of natural vibrations
normed in conformity vdth eq.(8.27);
Yi = mass characteristic of the blade during vibrations with respect to
the i-th harmonic;
yi
2m,*9 '
(8.35)
Uh
m
,«<i
equivalent mass of iDlade during vi'brations with respect to the
mode of the I-th harmonic:
w/?=J"/7i[^(')]2^r,
In the particular case when 1 = and when we can set y^°^ = r , the expression
for Yo "will coincide with the conventional expression (7 •49) for the mass char-
acteristic of a rigid blade [eq.(7»49)]-
pj = frequency of natural blade vibrations with respect to the mode of
the j-th harmonic; in performing the calculations, the value of the
frequency is attributed to the expression
Pj
2_0_
j£/l(/)'P^r-l-j;v[(/)Trfr
(S-36)
jm[/12
dr
0) = angular velocity (or rotor rpm, depending on the units in which pj
is determined); the ratio pj/uo should be dimensionless.
Other quantities entering eqs#(8.34) have the following values:
D„^\B'''7y'-y'dr,
H„=\B^''ry'd-r.
(8.37)
12m.
(8.37)
115
Here,
i^Ci) ^
equivalent "blade chord determined "by eqs.(8.3l) and thus
having different values depending on the niain"ber of the equa-
tion:
fi(0 = ^^ at / evenj
B<^>='B^ at i odd;
—J
7
and j^ = natural vibration modes of the blade whose harmonic is de-
termined by the value of the indices J and I.
The symbols of these modes are marked by a vinculum. This means that they
are normed such that y^ = 1. At the same time, the values of the first deriva-
tive of these modes, following differentiation over the blade radius 3^, are not
marked by a vinculum. This means that these derivatives are taken from the vi-
bration modes y^ normed such that y^ = R.
Equations (8.34) permit determining the vincuH of the coefficients K, L,
M, N, R, Q, S, T, U, L, and K for any zone of Table 1.8, if the coordinates of
this zone J and I and the number of the equation in the zone i are prescribed.
Thus, assigning in sequence the different values of J, I, and i and making use
of the operator which includes the operation prescribed by eqs.(8.34)> we can
determine all coefficients of the left-hand side of the equations in Table 1.8.
13. Determination of the Coefficients on the Right-Hand Side
of the Equation of Table 1*8
To determine the coefficients on the right-hand side ^^ and §^, it is neces-
sary to derive a special operator for the conputational program, in which these
coefficients are determined by the following formulas:
0,= _ix^^e, + ^r<7/;a„^; + ^I+^i^2C? + 0o;
0,=
>l;+-^^^C,
02 + <^i;
0j== -^>l,-|-Aj,2C^j O,-i-^^tanaj::j + 2^B} + 0\;
0, = ^5A-^t.2Cj + 0^;
^3 = -fi-'CA + <^3-
(8.38)
For harmonics above the third (at n > 3), we have
116
In eqs.(8.38), the new notations are used for a series of integrals;
(8.39)
Here,
where
?r = 9r«i;+A?,
cpr = constant coirponent of "blade setting angle, calculated from the
plane of rotation of the rotor; over the "blade radius, this quan-
tity changes only due to its geometric twist Acp.
The values of f^ are entered in the extreme right-hand column of Table 1.7.
Thus, eqs.(S.38) permit determining all coefficients of the right-hand side
of the equations if the value of I is prescribed.
14. System of Equations after Substitution of Eqs.(8.34) and (8.38)
If the values of the coefficients determined from eqs.(8.34) and (8.38) are
substituted into the equations of Table 1.8, then this same system of equations
can be represented in the form of Table I.9.
For simplicity, we limited oiirselves here to the case in which the calcula-
tion is performed with an accuracy to two overtones of vibrations and four har-
monics of rotor rpm. However, the above-derived mathematical formulas (8.34)
and (8.38) are written in a general form and permit calculation to ar^ desired
accuracy .
After evaluating the practical requirements, it becomes possible, in setting
vp the program, to limit the calculation to consideration of only four overtones
of natural vibrations and to six or eight harmonics of the rotor rpm.
117
TABLE 1.9
1
■ \
J-r«
*>ii%! \'»Jh'iM't)-htiif%
i^
T
itAfvw
n-
-ipfm^i
'H
Uma
iP^'m
k-U^,
t^v*J
iA-^
•««•
if^t^
•=Vv
ifi*MA :-tM'*«
Wi/r^
^
I^IW'I
i>^
•Hvw
^vv.
1^1^:^.*^%
^iAJ
•i/'^'W
^>'«
if'V^
V^A
WiViU
iV^
>**w
|Wi>S.-4J
-/Ml -it^^^^
<f«»ft
iW^vM
i*M.
{Ww»-»W
■^if^
uw
iW^^SJ
•^»##'
>*l^f-
•iz-S
^A'li^f.
-?^%
"^*.*
•*JU
^»W
/*v»
iWi'.^W
♦V*i«»
7/''**'*
?^***
IF
'n»
T>^«*'W
^/«V,
^/^'J^
iW^ir-W
■^*i^^
i^'v -i^^i^'y
-^«.*
-i/*'%
i^'-i'V
jHi^-*J y^'*«
5>t'j«>f,
-^^t'"*!
?>•«;»'#
-Mj» I^^'W-'J ->"^
-i#sf ; ->Mf,i, 4Mi*w ^j» "■'A^i^j A*'« if^vm-cnii i/^^^m va'*'^ 1 « -M*
^tjiV>j>''fV
^*f*
-fW*W*'J
-i^%
IM'»*ff/
-/"r#*
I^/^VW
V^iA;
/»«.«,
i^'<toj
i>^«w-i*j
fF
?/*'».f.
fw*^--w
■^/%
-/**^'/
rWv*-W
-«4vj>**c.J
■»»»
'^*,^j^'c,)
-ff rf.
I
iA(Fw'*v) [ -/«A
■«A*f^^4^
-^•^v*.
i-»«-?>'''b
-«A*I^
j^**.tv -iM%
"T
TMftrm-Owi 'J^'^'h -»V«if, I -f;i»*»
^M« if^faW-W ?>«**• ^/i'*,*,'
7>f'-«-#j| /tJM. jW^*iA,l -ii» T>*fiVtf»j> -^x.i, ^A'«.<i -?^W
?>t'«.f, ! ^)it'''.
4^(m,tC^ ^«,it WVrA^ -JJI, jM*^w6w) 'J^*t,9i
-?/***• i y>'*«fC#
M'
-/»»*
-i/^.t
-A(*f'# 7iMi^^*f0) «• -x^vIa^^' z**!* iW*vffJ
^^»*,r, {/i»#, . Xi^**^ ^x,i, -x/vf/V ""•
4>i»> ^/i*«,e, -/ii,,j, -f^ifjj^Hf^ «, -*ifvi/^*<9 *
jW^-y- -/t«,i, y/i^g, I -j/i'w„
'^{'■^h,!-*'-5V*i» liW«^-6«> ->tM, ^;t*x,e, "-i^i/^„
■iM(^n-f„)^ fiM,M, l ifc^;^,;' -""; jj^^^^"-^ -AM, j/'^^^r, -i
■/^*''r rH'n^tn) ^h, J^,^;^^ ,/ /^^,», \{^i3F,rC„). j^^, jj^'^^^r \
-iik9, fttf^h
4WArt ^*
^M^
"fv^'<^A i***fitift
fttih
T
tf^C -f*
4''Vi^*|
»/
:^A*k
*/
■ i/**ft^
*?,
#;
H
...
-M*
^t/«.i,'
,W/*t,' ♦/
-HA^'
*f
■ -l*riM^c)$, /tV,c, i^: ♦/
i^^x,^, I i^'/^„ !-,vfff„>c„j , ^M,i, _^^;J>
.i/('#„ ; i/i**,f,
-/*«,*, -j/i{lF„*S„) 3D„
-Wff jf^hnrfit) 'l^*th
V^'"/, |^;*'*,c,
->z«,5, ,-i;.f3F„tCff)
-L
/*!,'.
-f^v
♦/
-f^it't
'*;
iM%»t
!♦/
if^%*, t
,
'♦/
■> ' ,A
i 1 .-^-'
1 ! i-i
^^ tg = tan
Iliilll
15* General Computational Scheme
The system of algebraic equations entered in Tatile 1.8, together with the
system of equations presented in Table 1.7, represents a conplex system of equa-
tions permitting a determination of all unknown quantities entering into it. A
determination of these unknowns and primarily of all coefficients of blade de-
formation constitutes the ultimate purpose of the method of calculation pre-
sented here.
Above (Sect.?), in the calculation of bending stresses based on Galerkin's
method, we used various sinplifications in deriving the equations presented /109
here and in solving them. The use of digital computers greatly facilitates solv-
ing this system of equations without additional assunptions; this greatly in-
creases the reliability of the results. In ar^ case, the conputational errors
can be ascribed solely to the initial assxmptions. Perfonnance of all necessary
mathematical operations will introduce no errors and can be carried out at any
prescribed accuracy.
How does one solve this rather conplex system of equations? No doubt, the
sinplest method here is the method of successive approxmations in the form in
which it will be presented below. This method was used in programming and has
been checked by n-umerous calculations. The method converges rapidly and already
three or four approximations suffice for obtaining the necessary accuracy.
In applying the method of successive approximations, the unknown coeffi-
cients of deformation are determined in the sequence of ascending indices char-
acterizing their relation to the corresponding harmonic. Therefore, before pass-
ing to a description of the sequence of operations in the method of successive
approximations, a brief review of the determination of deformation coefficients
is required.
16. Determination of Defoirmation Coefficients
In determining the coefficients of def02?mation, the same principle is used
in all cases, involving the following. The deformation coefficients are deter-
mined in pairs from two equations of the system of Table 1.8 pertaining to the
cosinusoidal and sinusoidal coiiponents of some harmonic. The equations are first
transformed in the following manner: Determine the sum of the products of Af^
and aI^i coefficients in Table 1.8 as well as the deformation coefficients de-
rived before performing this operation, with the exception of the products con-
taining the coefficients outlined by a broken line in Table 1.8. These sums of
the products are transferred to the right-hand side of the equations. After
this, the deformation coefficients pertaining to lake harmonics (of cotirse, only
in the case of n > 1) and to the deformation modes are determined from two alge-
braic equations of the following form:
118
where
<Pn = 0^-A0,; 0^ = 0„-A0„.
The coefficients a^ and "b^ here have a generalized character in that such
a notation of the equations is possil^le also with the coefficients Cj^ and dj^; e^
and f^; g^ and h^.
The coefficients Q, S, R, and Q entering eqs.(8.40) are determined "by
ec^.(8*34) for the case of J = I# The value of i is even for the first equation
and odd for the second. Consequently, the coefficients, in this particular /IIP
case, can be written in accordance with eqs.(8.34) in the form
«2
S^-nDy,
R^nDj.
-^j[Aj + J-\^^Cj
(B,41)
In this case, at J = I, we have
Dj=-^B^')ry]^dr.
Neglecting the second term in the first equation of the system (8 .41) and
approximately setting B^^ = B^^ = t^. , eqs.(8.40) can he transformed into the form
used in the sinplified methods of calculation (see Sect.?)*
In fact, when making these assuir^Dtions, we multiply all terms of eq«(S.40)
hy vj
0)
Pj
Let us now introduce new notations. The quantity nj, determined "by the ex-
pression
".-ff/.o,.
(8.42)
will he called the relative coefficient of aerodynamic dairping#
The quantity v^ = ncu will he called the frequency of excitation of forced
vibrations .
Then eq.(8.40) can he rewritten in the form used in Section 7:
119
^'^"■+[(^)'-'l'"=^v.^..
Pj
From these equations, the anplitiide of the coefficients of deformation cor-
responding to the J-th harmonic of natural viT^rations can "be determined as
where
Xdyn ~ coefficient of dynamic increase in ajiplitude;
6gt^ - deflection of the "blade with respect to the mode of the J-th har-
monic under static application of external forces:
Pj
/[(^)-']"+-'ft)'
(8.A4)
Thus, the adopted form of determining the deformation coefficients theo-
retically coincides with the form used in prol^lems of mechanics when determining
the vibration anpHtude of a danped system, as described a'bove in Section 7*
It should be noted that, in determining the coefficients of deformation, /IJ
the equations of the system (see Table 1,8) are transformed into the form of
eq»(8*40) only at n > 1# In determining the coefficients relative to the first
harmonic, certain additional coefficients K, L, and U will enter the left-hand
side of eqs.(8./;.0) which, however, changes nothing in the essential aspect of the
matter .
A determination of the coefficients ao, Cq, Gq and go which define the con-
stant corrponent of the deformations proves to differ somewhat. These coeffi-
cients can be deteraiined for one equation with the number i = 0. However, so as
not to disript the generality of the approach, it is preferable to determine them
also in a program of two equations with the numbers i == and i - 1 whose coef-
ficients are determined by the same formulas [eqs#(8.34) J • In so doing, it is
necessary to put $o = 0. Such an approach yields a slight sinplifi cation of
the cocputational program.
17. Computational Program
In programming the calculation, the following sequence of performing the
necessary operations is used:
1. The natural vibration modes and frequencies of a blade in the thrust
plane are determined from a separate program which is absolutely necessary in de-
signing blades and thus must be formulated. The following quantities should be
120
defined for cairrylng out this calculation: j'^ , g^, a'^, and m®*^ . Here, a^^^ is
the distrilDution of "bending stresses over the "blade radius during its vibrations
•with respect to the normed modes of the J-th hannonic-
2. The parameters characterizing the flight regime of the helicopter are
prescribed: p., p, ci), a^ot^ Trcai ^ ^i> ^s* Here, oi^^^ and cpj-eai can be deter-
mined from calculation if the required propulsive force and thrust of the rotor
are prescribed. The cyclic pitch control angles 0i and 63 can be determined if
the required moments M^ and M^ due to the rotor blades and acting on the hub are
determined from the conditions of helicopter balancing. These operations are
usually included into the programming .
3. To arrive at the solution of the system of equations entered here in
Tables 1.7 and 1.8, it is necessary to determine the coefficients Yj and Hj.
This system of equations is solved by the method of successive approximations
where, in each approximation, all unknowns are determined in the sequence given
in Table l.lO. First, the coefficients in the first row are determined, then
those in the second row, and so on.
4. After determining all quantities given in Table 1.10, the parameters of
the flight regine cpreai > ck^o^. , 0-l, Gg can be refined and the calculation of all
coefficients can be carried out in the next approximation in the same sequence.
5- The sequenc of operations, indicated in Table l.lO, is repeated until
the difference of HKe deformation coefficients in two successive approximations
is less than the prescribed accuracy of calculating e6. The value of e6 can be
taken as equal to l/lOOO or somewhat smaller.
6- The magnitude of the bending stresses in the blade at each azimuth can
be detennined from the formula
where the values of 6j are determined from the deformation coefficients a©, aj^,
^n^ Co, Cn^ ^nj etc., in conformity with eqs.(8.27) and (8.28).
This sequence of operations constitutes the principle of •*-he method of cal-
culation presented here. Performance of these calculation pe • jits obtaining:
bending stresses and form of blstde deformation at each rotor azimuth, /113
with simultaneous determination of all harmonic conponents of these quan-
tities;
field of axial induced velocities in the plane of the rotor and all har-
monic ccnponents of this field;
angle of attack and blade setting angle in f Ught regimes with pre-
scribed values of propulsive force and thrust;
angles of deflection of the pitch control swashplate, necessary for cre-
ating the mcsnenbs M^ and M^ required for helicopter balancing.
121
18. Comparison of Calciilation with Experiment at Lavf Flyi ng Speed
At low flying speeds, the variat>le stresses measured in the blade are usu-
ally highly unsta*ble.
During a single flight regime flown Isy
WB.J fluctuate in magnitude by a factor of 2
s
Determined from
the Formulas of
Table 1.7
s
s
-a
s
s
S
s?
Total
Induced
Velocities
S
s
Partial
Induced
Velocities
n} u m
C
-o
§
*
S
K
§
■■1
Harmonic
Components
of Load
.5 ^^
C o «
.c?
.1^
icC7
•^
..s-
o
sg
■«§
4) .
C CO
•Si
^
^
^
<
•^
«?
i?
^
»?
^
■^
i
0?
CI
^
^
*
•^
^
«
CO O
o
•H
CO o
o
C O
8E
O
■sg
^1
■S'3
11
J3
o
■5 §
m g
C/3 cd
one pilot, the vibration ajrplitude
- 3* This can be attributed to the
fact that the angle of attack
of the rotor and the flying
speed in these regimes are ex-
tremely difficult to keep con-
stant. The fUght mode changes
continuously • However, the
designer is mainly interested
in the maxunum variable stress
anpHtudes, since these gener-
ally carry the greatest risk
with respect to fatigue in the
struct tire.
Usually, the maximum vari-
able blade stresses arise in
flight regimes with the largest
angles of attack of the rotor.
These regimes include braking
and steep descent at high ver-
tical speed.
To conpare the resxolts of
calculation and experiment,
one proceeds in the following
manner: Check all fHght
regimes with abrupt braking of
the helicopter before landing,
in which the blade stresses
were measured. From each
flight, select the maximum
(over the blade radius) anpH-
tude of stresses set xxp during
the entire landing mode. The
field of values of these
stresses is hatched in Fig. 1.33-
Then, calculate the
stresses for regimes with dif-
ferent flying speeds and with
an identical rotor angle of at-
tack. The flying speed in the
given regime will be character-
ized by p.. The results of
these calculations are given
in Fig. 1.33. The soUd lines
122
show the dependence of the calculated ina.yim\3m variable "blade stresses on flying
speed. Here, we examined regimes with an angle of attack a ~ 0, oi = JO^ which
can be achieved in a regime of abrij^jt braking, as well as with an angle of at-
tack a = 50° which is possible during steep descent at high vertical speed. The
dashed line shows the same dependence for a = and a = -6*^, but without con-
sideration of the variable induced velocity field. In performing these calcula-
tions, we investigated flight re-
gimes without overload when the
kgfmm^
Fig. 1.33 Results of Calculating Vari-
able Stresses, with Consideration of a
Nonuniform Induced Velocity Field and
Corrparison with Experijnent •
thrust of the rotor was equal to
the weight.
It follows from these calcu-
lations, that the greatest increase
in variable stresses at low speeds
is observed in flight regmes in
which the free vortex sheet shed
by the blades becomes two-dijnen-
sional. When the sheet is farther
removed from the rotor plane, the
variable stresses decrease greatly
and approach, in magnitude, the
stresses calculated without con-
sideration of the variable induced
velocity field.
A comparison of regimes with
identical angles of attack shows
a marked increase in variable
stresses, in a very narrow range
of flying speeds.
The results of the calcula-
tion reflect to some degree the
pattern of the phenomenon observed
in flight. Thus, as in flight,
the calculated values of variable
stresses increase at low speeds /114
and rise with an increase in rotor
angle of attack. However, there
is a considerable discrepancy be-
tween calculation and e^q^eriment.
1. At identical flight regimes,
the variable stress atrplitudes obtained from calculation were found to be lower
than those measured in flight tests.
2. The variable stress anplitudes obtained in calculation and e:5qDeriment
are quantitatively similar when conparing regimes with different angles of at-
tack, using, in the calculation, rotor angles of attack somewhat greater than
those occurring in flight.
3. A coir5)arison of flight regimes in which the stress magnitudes obtained
123
in calculation and experiment coincide shows a substantial difference in their
harmonic coiiposition# The content of high harmonics is greater in stresses
measured in fUght than in calculation. Thus, harmonics from, the foiarth to the
sixth predominate in stresses measured in an alDrtpt "braking regime, which are
shown in Fig.l.33» ^t the same time, stresses of the first, third, and fifth
haiTnonic predominate in varialDle stresses obtained l^y calculation. Here, they
are Usted in the sequence of descending am-
plitude. As an exajiple. Fig. 1.34 shows the
^^9/^^ — . — . , — . — , — ___. — , distribution of stresses over the "blade radius
and their harmonic content in a f Ughl regime
at Q^ = 50° and [i = 0.04S*
It should "be noted that, in the calcu- /115
lation, we investigated a blade with charac-
teristics ensuring the absence of resonance
at the operating ipm. Its resonance diagram
is shown in I^g.l.35» The operatir^g ipm
adopted in the calculation is shown on the
resonance diagram by a vertical line.
The presented data show that application
of the method of calculation, with considera-
tion of moderate induced velocities under the
same assurrptions as described in Subsection 3^
approximates the results of calculation and
experiment at low flying speeds. However,
further refinements are necessary to obtain
results useful for practical purposes.
Oj,G^
/
"N.
J'
5
'
/
y
\
r^
\
V
y
<?4
[N
/
\
^
r
-—
M
^^
L
- — '
^
rm
19. Comparison of Calculation with Experiment
at Moderate-Speed Mode
Fig. 1.34 Distribution of
Stresses over the Blade Radius
and their Harmonic Content at
Flight Regimes (y, = 0.048 and
oi = 50°).
Here, by moderate flying speeds we mean
all speeds at which the nonlinearity in the
relation Cy = f(cy) and the phenomena associ-
ated with flow conpressibility still have no
effect. In many cases, therefore, "moderate
flying speeds" conprise the crxoising speed
of a helicopter; this is especially of interest from the viewpoint of fatigue
strength, since the helicopter operates most of the ti^ne at this speed.
Figure I.36 gives a conparison of the airplitudes of variable stresses and
their first and second harmonics relative to the rotor rpm, obtained in calcula-
tions with stresses measured in the blade at cruising speed for |ji = 0.25 • The
stresses obtained in flight are shown by dots. The dashed line shows stresses
calculated with consideration of the assunption that \ = Xo^v = const, and the
solid lines with consideration that \ = var.
It follows from this diagram that the results from calculation and e35)eri-
ment at cruising Speed differ substantially. The total anplitude of calculated
stresses amounts to no more than 80% of the values measured in flight. This dis-
124
crepancy occiors mainly as a consequence of the difference in the values of the
second harmonic of the stresses relative to the rotor rpm. The coincidence in
the first harmonic of the stresses is rather good. The higher harmonics of
stresses in this flight regime are quite small and have no substantial effect on
the stress anplLtude.
results presented in Fig ,1.36 are typical for flight regimes with ij. =
= 0.25 and are duplicated on
almost all helicopters.
We also see from Fig.l.36
that consideration of variable
induced velocities, in this re-
gime, yields no noticeable re-
finement in the values of the
variable stresses* However,
when having to do only with /116
one hannonic - for exanple, the
fourth - it will be found that
its value increases greatly
when allav\rance is made for the
variable induced velocity field.
Therefore, such refinement is
highly inportant if this har-
monic is present and determines
the magnitude of forces acting
on the fioselage and causing it
1st harmonic to vibrate.
mo —
6th harmonic
5th harmonic
Ath harmonic
3rd harmonic
2nd harmonic
njrpm
Fig. 1.35 Resonance Diagram of Blade.
Above, we have said noth-
ing on the constant coirponent
of the bending stresses. Gener-
ally their magnitude, obtained
on the basis of calculation,
proves to be so acciorate that
it usually is not even measured
in flight. Calculation yields more reliable results in this case#
20. Possibilities_of Further Refinement of Calculation Results
As follows from the above, a calculation of variable blade stresses still
yields no results that could be conpletely satisfactory to the designer. If, at
moderate flying speeds, the results of calculation more or less satisfactorily
agree with experiment (althoiogh further refinement of the values of the second
harmonic is extremely desirable), a rather remote coincidence is observed at low
flying speeds.
In this connection, it is highly important to establish the direction in
which further refinement of the results is soioght. We can propose the following
in this respect.
125
In calculating variable stresses at low fljo-ng speeds, the most inportant
refinements conprise:
consideration of the effect of intrinsic induced velocities (atandonment
of the "steady-flow hypothesis");
use of the vortex theory which takes into account defonrations of the 7117
free vortex system (albandonment of the assunption that vortices are shed
from a rotor at a constant speed equal to the average disk downwash Xoav)*
*^i kgfmm^
fi} kg/mm*
•
• <
'
__
Variable
. striae
^ i
ami
olitude 1
-\/
--
/J
%
\.
*
//
/
s;
^
, )[/
^
f
K-var
\
^.
I
A=.
\
/
—
i
/
•""
1
\
t
.... - . ^
First harmonic
A
x\
VN
/ ,
/
\/^
N\
/
N
^.
~1/^ ^
;i=r.
-'
>
^^-:^
!
V
A=var\ >
^ j
If
1
1
\
<Sg /eg/mm* \ TV
c-*
kg/mm^
Fourth harmonic
1
. /
A^conjt
xTj
^-\
__j>
■^xC
L=
s
0,1 0,Z 0.3 OA 0,5 OJ OJ 0,8 0,9 f 0-1 0,1 0.3 0.^ 0.5 0.6 OJ Od 0,9 r
Fig. 1.36 Comparison of the Values of Varialble Stresses,
Calculated under Consideration of a VarialDle Induced
Velocity i^eld, with Stresses Measured in Flight.
In calculating varia^ble stresses at moderate flying speeds, where the main
discrepancy is observed in values of the second harmonic of the stresses, appli-
cation of the vortex theory for a finite number of blades and introduction into
the calculation of the effect of both extrinsic and intrinsic induced velocities
would constitute a highly useful refinement.
In cases of a blade of low rigidity in torsion or of excitation by the ex-
ternal forces of a rotor vibration mode coinciding with the flutter mode at a
frequency close to the frequency of flutter, a consideration of torsional blade
defonnations may yield noticeable refinements. The method of such a calculation
was presented in Section 7, Chapter IV of Vol.1.
126
Often, in calculating varial^le stresses at cruising speed (just as at maxi-
mum speed), a consideration of the nonlinear relations Cy = f(cy) and of flow com-
pressibility may lead to sutistanfcial refinement, a point to "be discussed further
in the next Section.
Section 9* Calculat ion of Blade Bending Stresses, -with Consideration /113
of the NonHnear Dependence of Aerodynamic Coefficients
on Profile Angle of Attack and Mach N-umlper
!• Flight Regimes.
Consideration of the nonlinear dependence of aerodynamic coefficients on
the profile angle of attack is necessary in fUght regimes in which these angles
attain such significant values that it no longer is possible to use linear de-
pendence [eq«(8.l)]* Such regimes pertain to flights at speeds close to maximum
and to low-speed modes in which, as a consequence of high blade loading and ex-
cessive nonuniformity of the induced velocity field at individual segments of
the disk area, the angles of attack enter the nonlinear domain of the dependence
Cy = f(ck')« In a number of cases, consideration of these nonlinearities is neces-
sary also in other regimes, including the cruising-speed mode.
In general, consideration of phenomena associated with flow compressibility
is necessary at high flying speeds for helicopters having rotors with high blade
tip speeds.
2* Dete_ra anatipn of Aerodynamic Loads
In Section 8, we had stipulated that the inflow angle to the blade profile i
is a small quantity; therefore, the approximate equation (8.23) was used in de-
termining this angle. Here, we stipulate that the angle $ can vary within limits
of 360°; therefore, its magnitude will be calculated by means of the formula
^x (9.1)
where the values of U^ and Uy are determined loy the expressions
L^_^ = a)/?(r-j-jj-sinO); ^
Uy = ii>R (x~~^^ cos ^? -~ y). I (9*2)
Equations (9*2) coincide with the formulas used in Section 8. This means
that, in their derivation, it was assumed that blade displacements are small so
that we can put
cos I
127
The value of the ai^gle § determined from eq.(9»l), when carrying out the
calculation on a digital conputer, is usually read out only in the range =F90*^*
This must "be taken into account in calculating the angle of attack "by means of
the formula
a=9-
-O.
Therefore, we can use eq.(9.l) only at U^ > 0.
from Pig. 1.37, we have
(9.4)
If Ux < 0, then, as follows
(9-5)
-90° <
The inflow angle deteraiined Iby eqs.(9.l) and (9»5) varies in the range
' < 270°.
<?<7r
Pig. 1.37 Diagram of Flow Past a Profile
for Determining the Inflow Angle § .
pressure -
If we assume that the /119
blade setting can be changed from
cp = -15° to 9 = +45°, then the
aerodynamic coefficients should
be prescribed within limits of
the variation of the angle of
attack from -105"^ to +315° .
The Mach number needed for
determining the aerodynamic co-
efficients is calculated by
means of the formula
M=^.
(9.6)
Here, ago is the velocity
of sound:
°"=/t-
(9.7)
where k is the adiabatic ex-
ponent and p is the atmospheric
The aerodynamic coefficients required for the calculation are determined on
the basis of wind-tunnel tests with the profile exposed to a circular air stream.
In coirputer calculations, the program compiled by engineer M.N.Tishchenko for
determining the aerodynamic coefficients is highly useful. In this program, the
effect of the Mach number M on the aerodynamic coefficients is taken into ac-
cotmt only in the range of p3?of ile angles of attack from ot = -2° to a = +15° •
128
In the remaining range of angle-of -attack variation, the aerodynamic coeffi-
cients are considered as independent of K.
The dependence of the lift coefficient Cy on the angle of attack a for the
profile NACA-230, which was adopted in one of the versions of this program, is
shown in Fig.l#38 as a typical exarrple.
^^h
-".^.^h
XT
t I
rtr
y+
0,6
1
1 1
1
-^ i ' '
=^ ^-^J 1
1 — .
1
1 j I
~ - -~^ — -^ 1 .
— k
~- ; . ^ \ ^ r-!
_^ ^ ^_^
-^^N — —
— M
\
1
\!
! 1
V ^ 1^^^ ,
-^4=l=N=i
^^
ZIX
\Zone allowing for the -
^dependence of cy on M
Hi I 1 1 1 j i '! 1 ; 1 ! I ! ? "L I
90^
no 150 130 2W 2^0 270 a
^(
n^
Fig«1.3B Dependence Cy = f(cy, M) Adopted in the Program.
If the lift coefficient Cy and the drag coefficient c^ are known, then the
aerodynamic forces acting in the flapping plane T and in the plane of rotation Q
can "be determined from the f omiulas
(9.8)
3« Method o f Blade Calculation as a System whose Motion
is Coupled "bv Prescribed Vibration Modes
As above in Section 8, the calculation of elastic blade vibrations reduces
to solving the differential equation
[Ely'Y-Wy'Y + my^T,
(9.9)
where, with the adopted assimptions, the aerodynamic force T is a nonlinear func-
tion of the displacements of the blade elements y.
In this case, it is convenient in solving eq.(9*9) to use a method where
the blade motion in time is found by numerical integration of ordinary differen-
tial equations obtained from eq.(9-9) by Galerkin^s met hod • In this approach to
the problem, these equations are coi^jled only over the aerodynamic forces. There-
fore, if - at some arbitrary time - the aerodjmaniic forces can be calculated.
129
then the blade deformations with respect to each vibration mode are determined
independently, provided these modes are orthogonal.
Let us represent the "blade vibration mode as the sum of a certain number of
natural vibration hannonics of the blade:
J
(9-10)
where
j = 0, 1, 2, •••* Jh (Jh Toeing the number of the higher overtone of
natural blade vibrations taken into account in the solution) ;
y^j^ = mode of the j-th overtone of natural blade vibrations normed such
that y^^^ = R at r - R;
6^= coefficients determining the magnitude of blade deformation with re-
spect to the j-th overtone.
As above, we will designate the coefficients 6j as the coefficients of /l2l
blade deformation. The values of 6 ^ are functions of time.
The coefficients of blade deformation 6j , in the present method of calcula-
tion, are taken as generalized coordinates of the system. Determination of the
law of their time-variance constitutes the content of the calculation.
After twice differentiating eq.(9«lO) with respect to time, we obtain
y=Zhy(^^; 1
J
(9.11)
On substituting eqs.(9*10) and (9»11) into eq.(9*9) and successively multi-
plying all terms of eq.(9.9) by y^^^ (where j = 0, 1, 2, ..., 3\i) ^.nd then in-
tegrating over the blade radius, eq.(9»9), ^y virtue of the orthogonality of the
natural vibration modes, will decoiipose into j^ + 1 independent equations of the
form
rrij'^l -{- CjOj = Aj,
(9.12)
Here,
C, = j E! [{yri' dr-^^N [{yJ)']^ dr ;
mj=^m{y^)^dr;
A;=^TyO)dr.
(9.13)
130
As mentioned above in SulDsections 1 and 2 of Section 7^ the quantities en-
tering eq#(9»l2) have a well-defined physical meaning. The quantity Cj, known
as the generalized "blade rigidity in deformation with respect to the mode of the
J-th overtone, represents also double the potential energy accumulated by the
"blade in "bending in a centrif-ugal force fieU with respect to the mode of the
same harmonic. The quantity m^ is the equivalent "blade mass reduced to its tip*
It is equal algo to double the kinetic energy of blade vibrations with respect
to the mode of the j-th overtone with a frequency p = 1. The integral A^ on the
right-hand side of eq#(9»l2) represents the generalized force and is equal to
double the work of aerodynamic forces in displacements caused by blade deforma-
tions at the j-th overtone.
It is known that the frequency of the j-th overtone of natural blade vibra-
tions can be determined from the formula
V'.
^'•=!/ %
Therefore, it is convenient to transform eqs»(9*l2), relating all terms to
values of m^ . These can then be written as
^I + PJ'/-J^ (9.14)
or
^
p)
^/i-/'?3=^. (9.15)
where 6^^^ is the coefficient of quasi-static blade deformations with respect to
the mode of the j-th overtone of aerodynamic forces T (see Sect. 7^ Subsect.7)»
As follows from eqs#(9»S) and (9 •2), the magnitude of the aerodynamic /122
force varies with respect to the blade azimuth and depends on the blade deforma-
tions or, more precisely, on the values of y and p determining the magnitude of
the relative flow velocity Uy. Therefore, for calculating the aerodynamic
forces, the values of y and p must be predetermined by means of
J
(9.16)
where P^^^ is the angle of rotation of the elastic blade axis relative to the
plane of rotation, corresponding to the normed natioral vibration mode of the i-th
overtone .
If the coefficients of deformation 6^ arid their first derivatives 6j per-
taining to some azimuthal blade position or to seme time t are known, the calcu-
lation can be performed in the following sequence ♦
131
First, determine the values of y and P from eqs.(9»16). After this, derive
the conponents of the relative flow velocity Uy and U^ as well as the velocity U
from ecp.(9*2):
U=^V'UIWI (9.17)
Of course, to determine the velocity Uy it is also necessaiy to know the
relative disk flow ratio X which, in the general case, is a variable changing
with respect to blade radius and azimuth. Determination of the quantity \ will
"be taken up in Subsection 5 of this Section.
If the velocities Uy and U^ are known, then the inflow angle § can be de-
termined from eqs.(9«l) and (9«5)* and the profile angle of attack a from
eq*(9.4)» The Mach num.ber is determined by eq.(9«6). These data suffice for
determining the aerodynamic coefficients for circular blowing of the profile and
hence for obtaining the aerodynamic forces T.
Thus, at the azimuth in question the blade deformation, rate of deformation,
and the aerodynamic forces T acting on the blade are known. Consequently, on
the basis of eq.(9»l^) it becomes possible to derive also the coefficients 6j
that determine the accelerations of the blade elements:
h
-~Pfi^ (9.15)
Next, by numerical integration of eqs.(9.14) with respect to time we can
determine the new values of the coefficients of blade deformation 6^ and their
first derivatives 6^ at the next blade azimuth after a certain time At, deter-
mined by the integration step . The change from the time t at which the coef fi-^
cients of deformation 6^ and their first derivatives 6^ and second derivatives 6j
are known to the next time t + At can be acconplished by various conventional
methods of numerical integration of equations.
As a typical exanple, we are giving the formulas for such a change, derived
from the Euler method:
The characteristics of various methods of numerical integration will be
discussed in greater detail below. In particular, it will be shown that the /123
Euler method represented by eqs.(9»19) is not suitable for calculating elastic
blade vibrations.
Numerical integration of eqs.(9«14) with respect to time permits determin-
ing the deformation coefficients and their first derivatives at a new blade azi-
muth. After determiniiig the new values of aerodynamic forces at this azimuth we
can also derive the new coefficients 6^. This process can be continued until
132
the coefficients of deformation are determined at all blade azimuths in one rota-
tion of the rotor.
If the initial values of the coefficients 6j and 6j are arbitrarily pre-
scribed, an integratipn of the equations over one rotor revolution will cause
the values of 6j and 6j , obtained by integration at the same azimuth, to differ
from the values taken arbitrarily at the initial time. However, if the blade
motion is stable, the numerical integration can be continued. Then, after sev-
eral revolutions of the rotor the motion will be established and will be re-
peated in each subsequent revolution. This steady motion is the sought solution
of eq.(9*9).
Thus, the method of calculation presented here is the solution of the Caucl:^
problem, with integration of the equations of motion of the blade with respect
to time at given initial conditions.
4* Mathem atical Formulas -for a B lade Model with Discrete Parameters
In practical calculations, a rotor blade is usually conceived as a weight-
less beam with attached concentrated loads simulating its mass. The aerodynamic
forces acting on the blade also can be conveniently represented as a series of
concentrated forces. Let us assume that aerodynamic forces are applied at the
attachment points of concentrated loads as though a separate flap with a certain
area S^ were attached to each load (see Sect.l, Subsect.9)» Then, the aerody-
namic forces can be determined by formulas analogous to eq.(9.8):
Qi-~{o,U^-CyUy^QS,U,, (9.21)
where the subscript i denotes all quantities pertaining to the blade section of
number i (see Fig.l.51)» The size of the area of the concentrated flap S^ is
determined by eq.(1.2).
For a rotor blade which is not represented as a beam with distributed para-
meters but as a model with a finite number of elastically coipled concentrated
masses, equations analogous to eq.(9»14) can be derived. However, the quanti-
ties mj and Aj entering the equations are not defined as integrals but as sums
of the form
^ [ (9-22)
i
where
mi = values of the concentrated mass of the system;
133
Ill III
j[^^ = values detemiining the nattiral vibration mode of the j-th over-
tone; here, the mode of natural vibrations should be represented
by a series of discrete values of the ordinates jj determining /l2U
the displacements of the i-th mass of the blade;
Ti = discrete values of aerodynamic forces determined by eq«(9«20)#
The calculation of a blade model with discrete parameters differs in no re-
spect from the calculation of a model with parameters continuously distributed
over the blade length. However, in digital conputer calculations it is much more
convenient to investigate a model with discrete parameters •
5# Consideration of a Variable Induced Velocity ELeld
%yplication of the calculation method presented above does not preclude the
possibility of corjsidering a variable, induced velocity field represented by the
relative disk flow ratio X in eq«(9»2). For this, in determinations of aero-
dynamic forces acting on the blade at the time t in question, the integrodiffer-
ential equation of the vortex rotor theory must be solved [see eq#(5»29) in
Sect. 5, Chapt.II of Vol.1].
Reduction of the problem of elastic vibrations of a blade to the Cauclny
problem, at determination of blade motion beginning with some initial time, leads
to appreciable sinplifications in solving the integrodifferential equations of
the vortex theory.
When the rotor advances one step with respect to azimuth, vortices that
have to do only with variations in circulation over the length of this particu-
lar step will be shed by the blade. All vortices shed from the blade at prior
instants of time are merely displaced in space but show no change in their cir-
culation. Therefore, in solving the integrodifferential equation pertaining to
some definite time, it is only necessary to find the relation between circula-
tion of the bound vortices and the vortices shed from the blade during its shift
after the last integration step. The magnitudes of circulation of all remaining
free vortices are already known in this case and are determined by the entire
history of the process of motion.
To sinpUfy the problem, we can take at the initial time some schematic
model of a vortex system consisting, for exanple, only of rotor vortices shed
from the blade tip with constant circulation over the length. It is assiomed
that, at the start of calculation, no free vortices can exist since the average
induced velocity through the rotor is equal to zero.
The method discussed here yields the maximum accuracy possible in the cal-
culation of induced velocities for a rotor scheme with a finite nimiber of blades.
However, lose of this scheme in other methods of calculation of elastic blade
vibrations will lead to serious conplications.
N-umerous difficulties are encountered in using calculation methods for in-
duced velocities based on a rotor scheme with an infinite number of blades, as
appHed to the method of calculation discussed in this Section. Thus, the method
of successive approximations generally appears siirplest. However, if we use a
134
method in which the induced velocities are calculated after conpleting the cal-
culation of "blade motion over each revolution of the rotor (when the values of
the aerodynamic forces T are known at all "blade azimuths and radii so that the
values of the circulation at the same point can be determined) and if we intro-
duce these velocities into the calculation of aerodynamic forces dioring the next
revolution of the rotor, it will be found that such a solution process does /125
not converge* Consequently, different methods "bypassing these difficulties must
be used; as a rule, this leads to appreciable complications which ultimately may
prove to be unwarranted.
6 • Characteristics of Numerical Integration of Differential
Ecfuatioris of Elastic Blade Vibrations
For a successful calculation of elastic blade vibrations, it is of inpor-
tance to select the most advantageous method of numerical integration, i.e., a
method of high accuracy and requiring a minim.um number of operations for solving
the differential equations of motion. Most of the machine time in calculation
is used for this operation. Its major portion is spent on determining the ex-
ternal forces. Therefore, the conputer time is determined mainly by the number
of times the equation of motion must be handled. This number is determined by
the chosen method and integration step. The smaller the step, the longer the
calculation.
An analysis shows that, when seeking a periodic solution of the problem of
elastic vibrations, the required integration step varies within very wide limits
depending on the type of numerical integration method used. Poor results are
obtained by many conventional nimierical integration methods, such as the above-
mentioned Euler method [see eqs.(9»l9)]» The well-known method of solution by
Taylor series was found to be just as unsuitable for the problem in question.
This method leads to the following fonnulas for the change-over from the time t
to the time t + At:
5/-fA/ = B^-pA^5^.
[ (9.23)
The value of ^t+Lt = ^(^t+At^ ^t+At) ^s determined from a differential equa-
tion. Here, At is the integration step.
The widely known Runge-Kutta and Adams numerical integration methods are
more suitable for the given case but still quite inconvenient.
The best method of checking the applicability of a given n-umerical integra-
tion method to the solution of the problem of blade vibrations is a numerical
solution of the equation
S-h2/7B + o = sinv/f, {9 ^2k)
describing vibrations of some mechanical model representing a mass attached to
135
a spring with a danper (see Fig#1.39)*
The rotor blade can be conceived as a set of a certain number of such
models, of different natural frequencies and different dajiping coefficients cor-
responding to the frequencies and danping coefficients of different harmonics of
blade vibration.
At relatively small steps At, the use of Taylor series for an integration
of eq.(9.2^) leads to a solution representing a vibratory process whose airpli-
tude tends to some definite value differing from the exact analytic value by a
quantity of the calculation error. Vfi-th an increase in the integration step,
the solution diverges at certain definite values of At. If the solution does not
diverge, the greatest error arises in resonance, i.e., at v = 1. Therefore, /126
we will now estimate the error with respect to this most severe case.
Fig. 1.39 Effect of Relative Integration Steps
on Accuracy of Solution.
Figure 1.39 shows the change in vibration anplitude values obtained as a
resxolt of the numerical solution, of eq.(9.24) "by means of Taylor series. The
exact analytic values of 60 and 60 were taken as the initial values. Cases with
relative danping coefficients equal to 2n = 0.1 and 2n = 0.2 and different in^
tegration steps were investigated.
136
The maximiain values of 6 ot>tained during the integration period with the
ordinal n-umber N were taken as the vibration auplitude A^ at this period and re-
ferred to the analytic value of anplitude
^^2^- (9.25)
It follows from Pi.g.1,39 that, during the nijinerical integration, the solu-
tion diverges from the exact analyt^ic curve. A steady vibratory process has an
anplitude always greater than the exact value* The larger the relative integra-
tion step AT, the greater will "be the error. Here, we will call the relative
integration step the quantity
^-T' (9-26)
where
At = integration step with respect to time;
T = vibration period of the model.
The magnitude of the relative danping coefficient n also noticeably af- /l27
fects the accuracy of the solution. It follows from the calculations that, to
obtain a satisfactory accuracy, the relative integration step should be of the
order of 1/200 of the oscillation period or even smaller.
In a nijmerical integration of equations describing elastic vibrations, it
is inportant not only to seciore the required accuracy but also to use an inte-
gration step in which there would be no divergent solution.
The determination of the limit step of integration, at which the solution
will still be stable, can be acconplished in the following manner:
Equations (9*23) and (9*24) can be regarded as some system of difference
equations. To determine the stability of the solution, we will discuss a homo-
geneous system of difference equations [without the right-hand side of eq.(9.24)]»
Equations (9*23) are written in a somewhat more general form, introducing
some constant coefficient k:
S/+A^ = S^ + AifS^ + xA/2B\ ]
Sf+A^ = 3, + A^5;. J (9.27)
At H = 0, these formulas coincide with the Euler equations (9.19) while,
when H = -J, they coincide with the Taylor equations (9 •23)*
From eq.(9.2^) for the case of sin vt = 0, we derive the value of 6^ ; sub-
stituting this into eq.(9.27), we obtain the following system of difference
equations:
137
6,+,,= _A^S^ + (l-2;rA0V J (9.28)
The solution of this system "will be sought in the form of
S^ = 5a«; S,+i^ = ^aC'^+i). 1 (9-29;
Substituting eq.(9«29) into the system of homogeneous difference equations
(9 •28), we obtain the characteristic equation relative to of. Prom this equation,
we find of :
a=l-A^ ^^+-L^^^j=p^^i/ ^^_1,_L^^^J2_|, (9.30)
To keep the values of 6^ from approaching infinity as n -♦ 00, the condition
lal<l
is necessary.
At relatively small At and n, the value of a - as follows from eq.(9*30) -
is a coup lex quantity.
After determining the modulus a, we obtain the condition of a nondivergent
solution:
\^J^\-^^fi~-2^tU^~%Lt\
<1
(9.31)
or
A^-^2(;.+^xA^j<0. (9,32)
Hence, /l28
A^<7^^. (9.33)
1 — %
If the integration step is related to the oscillation period of the system T
equal to 2rr in the examined simplified model, we obtain the 'condition of a non-
divergent solution
t™^ n (9.34)
138
Then, for the Euler method at h = 0, we find that the solution is possible
at
« (9.35)
and, for the Taylor method at k = -J-,
^■<— . (9.36)
TC
Thus, in order to avoid, a divergent solution, a step smaller "by a factor
of 2 is needed in the Euler method than in the Taylor method. Both methods give
a divergent solution no matter how small the integration step, provided that the
relative darrping coefficient n is equal to zero.
With an increase in n and At, the value of a becomes a real numlber. In
this case, the value of a can never l^e greater than unity but may be a negative
quantity greater than unity in absolute value.
The condition that or < 1 is observed if
(2x— l)A/2^4^A^f — 4>0. (9.37)
Hence, instability of the solution for the Euler method at h = will occur
at AT > '^^ — if and only if n > 1, whereas for the Taylor method (k = ^)
"" - 1
this happens at AT ^. However, these conditions are usually covered by the
2TTn
more rigorous condition (9 •36).
If these results are transferred to a system representing a rotor blade,
then the magnitude of the relative step must be selected on the basis of the
period of the highest harmonic of vibirations possible in the system, since this
will result in the smallest value of the required step at which numerical inte-
gration is possible.
Figure I.40 shows the typical character of variation in the natural vibra-
tion period of a blade T^ and in the relative coefficient of aerodynamic danp-
ing n with respect to the number of the harmonic of the vibration j. The value
of the vibration period is calculated in degrees with respect to the blade radi-
us. The same diagram shows the dependence of Jp^ on the number of the harmonic;
Pj is the frequency of the j-th overtone of natural blade vibrations calculated
in oscillations per minute. In the range of lower harmonics, the quantity pj
changes greatly with any variation in rotor ipm from n = to the operating rpm
n = nop •
If we limit ourselves in the calculation to a consideration of only the
first four harmonics of natural vibration, including the fimdamental, which
139
lillllllll
0.15
usually is sufficient for ol^taining the accuracy required in practice, then the
integration step must te selected on the tiasis of the period and coefficient of
relative danping of the highest harmonic of natural vi*bration, the third for /l^
this system.
If we assume that the vibration period with respect to the third harmonic
cannot iDe shorter than 45*^ with re-
spect to the rotor azimuth and
that the relative coefficient of
aerodynamic danping will not be
lower than n = 0.07, then - to ob-
tain a nondivergent solution - the
integration step in conformity with
eq*(9»36) should be less than 2°
and in conformity with eq.(9»35)
less than iP with respect to azi-
muth. The step would have to be
shortened much further to obtain
satisfactory accuracy (Fig.l.39)»
This exairple shows that an ap-
plication of the above integration
methods to blade calculations gives
unsatisfactory results. For this
particular exarrple, the Runge-Kutta
and Adams methods permit using an
integration step of the order of
3°, but they are not too suitable
since they require storage of an
excessive number of variables, cal-
culated for the preceding instants
of time, in the coirputer memory.
G-ood results are obtained by
a previously mentioned integration
method (Chapt.17,, Sect.? in Vol.l)
with expansion of the solution in
each integration step. This method is
problem in question and is being used
0.05
01Z3^S5 10
30 J
Fig. 1.40 Dependence of Vibration Period
T^j; of the Relative Coefficient of
Aerodynamic Danping n on the Number of
the NatiJiral Vibration Overtone j .
a Taylor series and with recalculation of
conpletely suitable in applications to the
at present in numerous conputer programs.
The change-over from the time t to the time t + At is acconplLshed by this
numerical integration method, in the following sequence:
First rough calculation:
130
5l+A/ = S^ -I- A/8^+— A^2s\
S<+A/^S^-{-A/6^.
140
t^At
Fig. 1.41 Dependence of the Variable 6
and its First and Second Derivatives
with Respect to Time.
IR^OJ
-^ur
Fig. 1.42 Results of Nixrnerical Solution of
Eq.(9#26) as a Function of the Relative
Integration Step •
Here, K^ + At = i^(^t+At * ^t + At ) is
detennined from a^^ differential
equation. Then, S^y is obtained
from the f omaula
-\KK
^t-^Mj'
(9.3B)
Recalculation:
SlU = S, + A/B, + ^A/2s^,;
Here, 6t+At =^i^t+^t, ^tVAt)
is determined from a differen-
tial equation.
The values of 6t"+At^ ^t+At*
and 6"^ At ^^® considered final
for the time t + At.
The change of variable 6
and its first and second deriva-
tives "with respect to time, de-
termined in conformity -with
eqs.(9»3B), is shown in Fig. 1.41*
Figure 1.42 gives the
steady solution of eq.(9*24) o^>•
tained as a resiJ.t of numerical
integration by this method. The
solution is given for different
values of the integration step.
The heavy Une shows the exact
analytic solution.
At a relative step of 1/72
and less, a numerical integra-
tion yields a solution almost ex-
actly coinciding with the analy-
tic solution. At a larger rela-
tive step, a substantial dif-
ference occurs between the exact
and numerical solution, which is
apparent from Fig. 1.42*
At a relative step of
l[,.l
Ar>—
(9.39)
the solution diverges.
To preclude the possil^ility of divergent solutions in the system, the in-
tegration step should not be greater than about 1/3 of the period of the highest
vibration harmonic of the system, which has the smallest period. An inportant
advantage of this method Hes in the fact that the Umit integration step is
practically independent of the magnitude of the relative danping coefficient.
A couparison of the limit steps Aiim ^^'^ "^^^ examined integration methods
as a function of the n-umher of the higher harmonic j^^ of natural vibration of
the system is shown in Fig. 1.43 for a blade with the parameters shown in the
diagram of Fig. 1. 40.
If we restrict ourselves to a consideration of only the first four har-
monics of natural vibration, then in conformity with eq.(9-39) it suffices to
have an integration step of about 15*^ with respect to the blade azimuth, i.e.,
by a factor of about 7 greater than in the same method without recalculation, in
order to obtain a nondivergent solution.
The results of solving eq.(9.24) permit an approximate determination of the
error in the auplitude values corresponding to different harmonics of blade vi-
bration as a function of the integration step used. By error, we mean here the
difference between the exact analjrfcic value of the vibration atrplitude and the
value obtained as a result of numerical integration. This difference is always
positive in integrations by means of a Taylor series with recursive calcula- /132
tion. This means that the numerical solution always leads to underestimating
the vibration ajrpHtude.
The calculation errors, in percentage of the exact values of the anplitude
for different blade vibration harmonics with ordinary parameters as a function
of the step used in an integration by Taylor series with recalculation are given
in Table 1.11.
TABLE 1.11
Number
of Overtone
Calculation Errors in Percentage of Exact Value
of Amplitude for Integration Step in Degrees
vr A ^H^ T \*» * ^^-i^^-^ ^^
0.3 1 1.0
2.5
5 1
10 20
Fundamental
<0,1%
<0.1%
<0.1%
0.3%
5%
25%
1st
<0,1%
<0.1%
0.4%
6%
12%
50%
2nd
<0.1%
0.3%
5%
25%
45%
80%
3rd
<0.1%
0.4%
15%
30%
75%
5 th
<0.1%
2%
20%
70%
10 th
1%
30%
90%
20 th
40%
Divergent solution
30 th
90%
142
7 ZJ
ro 20
Number of overtone
^^ Jh
Fig*l,43 Coirparison of Lmit Steps for
Two M-umerical Integration Methods.
Lunit step in integration with ex-
pansion of the solution in a
Taylor series;
limit step in integration with re-
calculation by eqs.(9*3S).
The presented data show that
the magnitude of the required inr-
tegration step and hence the cal-
culation time are determined
mainly by the parameters of the
system representing the rotor
blade. The more degrees of free-
dom the system has and the more
natural vibration harmonics it /133
possesses, the smaller will be
the -vibration period of the high-
est harmonic and the smaller
should be the integration step.
Therefore, the calculation time
is substantially shortened if the
ntunber of degrees of freedom of
the system is reduced. All these
considerations are especially im-
portant when using direct calcu-
lation methods which do not em-
ploy limitations inposed "upon the
modes of blade vibration. These
methods will be examined in Sec-
tion 10 of this Chapter.
Numerical Integration Method
Proposed by L.N.Grodko and
O.P.Bakhov
In the nimaerical integration of differential equations of elastic vibra-
tions of a blade by the method proposed by L.N.Grodko and O.P.Bakhov, the value
of the coefficient k in eqs.(9»27) is taken as equal to unity.
The stability condition (9*31) is siirplified and takes the form
!]/l— 2a//z|<1.
(9.40)
Consequently, at k = 1 there cannot be a divergent solution with a conplex
value of a . From the stipulation that a is a conplex number, the condition
(9.40) is valid only for values At :^ 2 - 2n.
From the condition (9»37) it follows that the solution cannot be divergent
as long as
A/<2
V 2x - 1 ' (2x
/l2
2n
(2x-l)2 2x-r
(9.41)
Hence, at h = 1, it follows that
143
IIIIIH^^
lllllllllll
^ ^Vl^rfi ^n
(9.42)
As in an integration "by a Taylor series "with double recalculation, this
method does not give a divergent solution at n = and has approximately the same
A/alue of the limit step.
Its accuracy -with respect to the solution of problems of elastic vibrations
is no worse than that for the preceding method. The volume of conputational
operations is cut almost in half. Therefore, this method of numerical integra-
tion can be recommended for practical use.
8, Sequence of Operations in Recalculation and Practic al _Evaluat_ion
of Different Integration Steps
As a whole, the calculation of elastic blade vibrations is carried out in
the following sequence:
1. Assign arbitrary initial values of 6j and 6^ at the azimuth \lr = 0.
2. From eq.(9.20), determine the magnitude of the aerodynamic forces Ti ,
for whose determination the following parameters should first be calculated: yi.
Pi. U^i. Uyi^ ^1^ Q'l. ^1^ Cyi^ ^^^ Cxi .
3. From eq.(9.1S), determine the values of 6j • The values of mj and pf
entering this equation are calculated beforehand after determining the natural
vibration modes of the blade and remain constant dioring the calculation.
4. The change-over to the next azimuth is accoirplished in conformity with /134
the selected numerical integration method, for exanple, by means of eqs.(9-3S).
8JV./-5, + a4,;
'i^^ Aj 2^11
(9-43)
The values of S^Vai* ^f+At> ^^ '^t+At ^^^ ^^^ time t + At are considered
final. For changing to the next azimuth, the entire cycle is repeated.
Iif4
This integration method can "be reconnuended as fairly exact and has "been
quite fully checked in practice in calculations of elastic blade vibrations #
The numerical integration is carried out over several rotations of the
rotor, until all values of 6^ in two successive revolutions differ by less than
the prescribed accuracy of the calculation. Calculations show that ar^ pre-
scribed accuracy can be achieved in this manner.
In practice, however, it is assumed that the calculation is conpleted as
soon as the accuracy of determining the deformation coefficients becomes equal
to R/lOOO (R being the rotor radius). If necessary, a greater accuracy can be
prescribed.
The magnitudes of bending stresses at each azimuth can be determined by the
formula
where a^ is the normed value of bending stresses, i.e., stresses during blade
bending with respect to the normed natural vibration mode of the j-th overtone.
The period of the process of transition to steady motion largely depends on
the assigned initial values of the deformation coefficients. At properly posed
initial values of 6j and 6j , the calculation is corrpleted after checking two
revolutions of the rotor. At poorly determined initial values of 6j , the calcu-
lation may drag out to 8 - 10 revolutions .
The possibility of refining the flight regime parameters 9o, Q'rot, ^> and
Xoav after checking each revolution should also be included in the calculation
program. The indicated parameters are refined such that the rotor produces the
magnitude of thrust and propulsive force prescribed in the initial data. Thus,
it is logical that the calculation time is determined also by the correctness of
prescribing the parameters of the flight regime.
To refine the flight regime parameters and also to solve other problems, /135
various integral rotor characteristics such as thrust %o%9 longitudinal force H,
torque M^., etc. should be determined during the calculation.
On the basis of practical requirements, blade vibrations can be represented
sufficiently conpletely by four natural vibration harmonics. In this case, even
considering Table 1.11 which shows that the largest errors arise in resonance,
satisfactory accuracy can be obtained at an integration step Ai|r = 2*5° •
However, for all practical piorposes in the absence of well-defined reso-
nance or in the presence of darrping forces in the system sufficient to produce
a danping coefficient greater than 2n == 0.1, the accuracy of the calculation
used in conpiling Table 1.11 is not entirely lost, even at a step A\lr = 5*^ or, at
times, even at a step Aijf = 10^. This fact is of great inportance in saving time
when using digital conputers of moderate speed. Thus, with the "Strela" conpu-
ter, only 6 min are required to determne the motion of the blade over one revo-
145
lution of the rotor, at an integration step of 10° • On decreasing the step, the
machine time increases greatly, rising so much at a step of 2.5*^ that performr-
ance of the calculation on this conputer t)ecomes difficult • These considera-
tions lose their meaning when the calculation is performed on the high-speed
M-20 conputer.
^0
0.10
0.05
QMS
^z
-QM5
^3
'0.00 J
/
^-^
■>
r
\
/
\
/
\
/
\
J
V
\
/
\
,/
/
\
v
90
180
no
f^
f 1
/
^^
\^
\
(f
1
^
^ .^
\
K
/
"^ Numerical
integration
\
method^ /
\
K
y
90 "
1
z=^SQ
2
70
*•
/
Numerical integration
B.C. (Jul erkm
method ,
/-
9
ISO
1
270
If
^
^
-s
Lj
a
21h ■ ^
1 ■■
—
-
mk
r
Fig.l.Zt4 Conparison of Deformation Coefficients Obtained
by Solving the Equations with Galerkin's Method and
with Numerical Integration for Cy = c^ a and iJi = 0.3*
As an exanple, iilg*l.Z^4 shows the values of the deformation coefficient cal-
culated for a helicopter in a flight regime with a speed corresponding to |jl =
= 0.3* For the helicopter under study, this regime is far from flow separation;
therefore, the calculation is performed in a linear setxp with the assimptions
described in Subsection 3 of Section 8. With these assutiptions, the calculation
was carried out at integration steps of 2*5*^, 5°, and 10°. To all intents and
purposes, the results of these calculations, shown in Fig.l.Z^i^ by a solid line,
coincide fully. On the basis of these data, it can be concluded that, in flight
regimes sufficiently remote from flow separation when the linear approach to the
•solution of the problem is used and low vibration harmonics prevail in the solu-
tion, at appreciable forces of aerodynajnic danping acting on the blade, the cal-
1M>
culation can be performed with an integration step A^
loss of acctiracy.
10^ without substantial
The picttire changes for regimes in which onset of flow separation occiirs.
Such flow separation leads to an increase in vibrations relative to higher har-
monics and to a sharp decrease in the coefficients of aerodynamic danping. As
a consequence, the integration step must
be shortened •
Figure 1.45 gives the conputa-
tional data for the danping coefficients
using the steps A^lr = 5° and A^lr = 10°
for the same rotor as above but in a
regime at p. = 0.4 with incipient flow
separation. The calculation was car-
ried out with consideration of the non-
linear dependence of the aerodynamic
coefficients on the profile angle of
attack a and on the Mach number M. Flow
separation leads to a pronounced in-
crease in the vibration airplitude with
respect to the modes of higher har-
monics which, as is known, even without
separation have lower aerodynainic danp-
ing coefficients. Therefore, a de-
crease in aerodynamic dairping at flow
separation primarily affects the vibra-
tion aiiplitudes with respect to these
modes. Due to this, calculation with
the step Ailr = 10° introduces substan-
tial errors into the calculation of the
deformation coefficients 63 and 63 .
In Fig, 1.45 this is illustrated on hand
of a comparison of the calculation, at
A^ = 5° • Therefore, to reduce the er-
ror in calculating deformations in re-
gimes with incipient flow separation
the integration step must be reduced to
values of the order of A^ = (2.5-5)° •
Fig. 1.45 Deformation Coefficients
at Incipient Flow Separation, for
jjL = 0.4*
9. Comparison of Results by Numerical
Integration Methods with Calcula-
tion of Harmonics
A method of stress calculation with respect to harmonics was presented
above in a linear arrangement, using the assunptions set forth in Subsection 3
of Section B. 'Such a method will be successful for flight regimes sufficiently
remote from flow separation. It has a number of advantages, the first being the
relatively short calculation time.
In Fig.l.Z|4 the deformation coefficients calculated by the harmonic method
147
<y t
Fig.l«Z|j6 Maxxmum iijiplitude of VarialDle
Stresses over the Blade, as a Function
of Flying Speed.
presented in Section 8 are shown "by
a "broken line, for conparison in
the same flight regimes at ijl =0-3
with a linear dependence Cy = Cy a.
A study of the calculation methods
shows satisfactory agreement of the
results. The slight difference can
"be attributed to some difference in
the initial parameters of the flight
regime •
10. Some Calculation Results
We will here present individual
results that characterize the new
possibilities for theoretical in-
vestigations offered ty the method
of numerical integration with con- /137
sideration of a nonlinear dependence
of the aerodynamic coefficients on
the angle of attack a and the Mach
number M, in conparison with linear
methods of calculation.
One of the major advantages of numerical integration is the possibility of
making stress analyses tinder conditions close to flow separation regimes.
Calculation shows that, on approach to flow* separation, the aerodynamic
danping of blade vibrations decreases steeply and the anpHtude of vibrations
having harmonics in resonance or close to resonance with the natural blade vi-
brations increases. A study of the deformation coefficients plotted in Fig. 1.45
indicates that vibrations at the first overtone occur mainly with the second har-
monic, those at the second overtone with the fourth, and those at the third over-
tone with the sixth harmonic to the rotor ipm, i.e., only with frequencies close
to the natioral vibration frequencies of the blade in question. An especially
pronounced increase in vibration aiiplitude takes place with respect to modes of
the relatively higher vibration overtones, as demonstrated in Fig. 1.45 on the
exairple of the coefficients 63 and 63.
The onset of flow separation is characterized by a marked increase in the
anplitude of the variable blade stresses. Figure 1.46 shows the values of maxi-
mum anpHtude of variable stresses over the blade radius as a function of fly- /139
ing speed, calculated with consideration of the Hnear and nonlinear dependence
Cy = f(c^, M). A marked increase in stresses is a highly useful criterion for
determining the onset of separation in calculating the aerodynamic characteris-
tics of a rotor.
The harmonic content of the variable stresses set ip during flow separa-
tion and their distribution over the blade radius are shown in Figs. 1.47 and
1.48.
148
19
V
16
^5
13
n
u
w
9
8
7
6
5
A
3
2
1
q.f 0.2_0.3 OA 0,5
L in ear
onlinear
calculation^
■ f
0.1 0.2 0.3 0.^ 0,5 0,6 0.7 O.g 0.9
Fig ,1.47 Distrit)ution of Variable
Stress Airplitudes and the Two First
Harmonic Stress Coirponents over the
Blade Radius at p, = 0.4*
3
z
1
a
7.
s
5
1 1
Non
line
ar
ion
Linear
calculat
1
calculation
-^/\^^^
\
^
■— "^^
i^jp-^-
-Km
r' 1
■^
^
1"^"
^-r ■-
\
o.f 0.2 0.3 a^ 0,5 as 0.7 \\0.9 p
1 I
'
Nonlinear
calculation^
J
,
w
\
1
'-CT
V
^,-
l\.
r
i
\
^^
\
t
I
— r
x^
Linear ^
calculation
\
^^
^
\
v_
r
1 — V
u^_U-M — -
i—
^ — "
"
L_^
0.1 0.2 0.3 0^ 0,5 0.6 0,7 0,8 0.9 r
1 r
\ 1
r--^
Linear
calculation
Nonl inear
calculation^^
1 — ^
1^
\
— L_IV-
' 1 T.-.r
\
0.1 0.1 0.3 0.U 0.5 0,6 0.7 0.8 0.9 r
' Nonlinear
calculation
\
Linear
calculation
J 1
\
^-
~•-.^
\
/*
^
^<
^
\
Q.1 O.l 0.3 QM 0.5 0.6 0.7 0.8 0,9 f
Fig. 1.48 DistriTDution of Anplitude
of the Third, Fourth, Fifth, and
Sixth Harmonic Components of Stresses
over the Blade Radius at |i = 0.4»
It should t)e noted that a substantial difference is also observed in the
results of linear and nonlinear calculations in regimes siofficiently remote from
flow separation.
Flgiire 1.49 gives the deformation coefficients calculated for the same
helicopter at ^jl - 0.3, with a linear and nonlinear dependence Cy = fCor, M);
Fig. 1.50 shows the corresponding harmonic conponents of stresses and their anpH-
tude a A constructed over the blade radius. As indicated bfy this diagram, the
results differ substantially.
Thus, already the few data presented here show that the calculation of vari-
able blade stresses with consideration of the nonlinear dependence Cy = f(cy, M)
yields a large number of interesting characteristics that have a substantial inr-
fluence on the rotor strength.
149
o
6f, kg/ mm
^3
0,1 0,1 0.3 OM 0,5 0,5 OJ 0,8 0.9
3rd harmonic iNonlinear calculatiqn___^'^^~-^
1='^==^^: f~ I iLinear calculation ^
^v, l^th harmonic
Nonlinear
'calculation~
*V J Wrtttrg= •■ i T i H-' It! 1 ■
Linear
]calculation~
-*f^
Fig. 1.49 Conparison of Deformation Coefficients
Calculated mth Consideration of the linear
and Nonlinear Dependence Cy = f(a, M) for the
Regime iJ> = 0.3 Far from Flow Separation.
Fig. 1.50 Distribution of Variable Stress
Anplitudes and the Four First Harmonic
Coirponents over the Blade Radius for \l = Of3.
lEi
Section 10. Calculation of Flexural ViTjrations mth Direct Detennination Z342
of the Path s of Motion of Poiribs of the Blade
1. Principle of. the Method of Calculation
In Sections 7, 8, and 9 we presented methods of calculating flextiral "blade
vibrations where the deformation mode was determined "by Galerkin's method. For
this purpose, the "blade deformations were expanded in a series in prescribed
known functions. As such functions, we proposed using the natural flexural vi-
bration modes of a "blade in vacuum. In this respect, it was stated that, for
practical puiposes, it is sufficient to limit the calculation to the first four
harmonics of natural vibrations.
Here, we will discuss methods that eliminate this assimption and permit a
determination of blade deformations by a direct calculation of the paths of mo-
tion of a certain number of points of the blade, without expansion of the vibra-
tion mode in known fiinctions^^.
To determine the motion of individual points of the blade, it is convenient
to use a blade model with discrete parameters. In this case, the mass of the
blade is simulated by several concentrated loads distributed over its length.
For such a mechanical model, we can derive a system of differential equa-
tions of the form
^/^/ = Q + 5^i,
(10.1)
where
^^i = 0, 1, 2, • - ., z;
Ji ~ second derivative with respect to time for displacements yi of the
i-th concentrated load with mass mi ; the values of ji are reckoned
from the plane of rotation of the rotor;
Gi = elastic force acting on the i-th mass mi by adjacent segments of
the mechanical blade model;
Ti = external aerodynamic force acting on the i-th point of the blade
where one of the concentrated loads is situated.
The system of equations (lO.l) describes the motion of all masses of the
mechanical blade model. Thus, it conprises equations with variables yj equal
in number to the masses of the mechanical model in question.
However, not all variables yj entering the system (lO.l) are independent,
since the motion should satisfy the condition of equilibrium of the entire
system:
(10.2)
^'' Such a method for calculating a helicopter blade was first used ty R,M.Zano-
zina.
151
It is preferable to consider that the displacements of all masses, except
for the root mass mo>are independent variables. Then, the motion of the root
mass, if we assume Tq = 0, can be determined in conformity with eq.(lO.l) as
^o^o=^o>
where
1
(10.3)
This condition of equilibriijm of forces is automatically satisfied when /US
using the formulas presented IdoIow.
Thus, the system in question can be described by independent variables y^
whose number is lower by one than the number of concentrated masses of the me-
chanical model. Consequently, the number of degrees of freedom of this system
is equal to the number of segments of the calculation scheme and is lower by one
than the number of concentrated masses.
El
Axis of flapping
hinge
\o\x \r Iz \3 U \5 \6\ ( 1^
^5 ^z-Z ^Z'l
Fig. 1.51 Blade Model Examined in the Calculation.
The solution of the system of equation (10.1) can be obtained by numerical
integration with respect to time. For this, it is necessary at each instant of
time to determine the forces Cj and T^ #• The forces Ti can be determined from
eqs.(9.20) whose derivation is given in Section 9» A determination of the elas-
tic forces Ci has many peculiarities, which we will discuss here at some length.
152
2. Determination of Elasti£ F orces Applied to a Point
o^ tile Blade Vy Ad.jacent Segments
Let lis make a more detailed analysis of the mechanical blade model used in
the calculation. First, let us exajiane a "beam^type model. We will represent
the blade as a weightless free beam governed by certain boundary conditions at
the ends and divided into z segments, along whose edges concentrated loads are
placed (Fig. 1.51). The lengths of the segments can be different.
As before, we represent the flexural rigidity of the blade as a stepped
curve so that it remains constant over each segment. We will assume the cen^
trifugal force as applied only
to the loads. Therefore, its
magnitude will remain constant
over each segment. We will
also assimie that the aerody-
namic forces are applied only
at the points of attachment of
the loads as if a separate
flap with an area Sj were at-
tached to each load.
To produce the conditions
of blade attachment at the
root, we will assume that the
centrifugal force is sensed by
a special attachment of root
mass mo, able to move freely
in vertical direction. When
solving this problem it is not
necessary to create freedom of
vertical motion of the root
mass. However, in other prob-
lems associated with a deter-
mination of synchronous vibra-
tion modes of the blade and fuselage, this condition is necessary. If the /II4U
fuselage vibrations are disregarded and the blade is considered as attached at
the hub on a rigid base, the conditions of root attachment in the calculation
are established by prescribing the necessary - usually rather large - mass mo*
It is logical that such an idealized scheme will yield a more accurate de-
scription of the real pattern of blade vibration the larger the number of seg-
ments into which the blade is divided. The blade can be represented with suf-
ficient accioracy by a scheme in the form of a beam consisting of 25 - 30 segments
and of the same number of concentrated loads.
To determine the elastic force Cj, we will construct the equations of blade
deformations. Figure 1*52 shows the forces acting on two adjacent segments of a
deformed blade. let us write out the equations of deformation of these segments.
Since the inertia and aerodynamic forces for the mechanical model in ques-
tion are appHed only along the edges of the segments, the deformations of each
Fig. 1.52 Polygon of Forces Acting on
Adjacent Blade Elements.
153
segment can be deteraiined "by the equation
[EIfr-[Ny^Y^Q, (10.4)
The magnitude of the flexural rigidity EI and the centrifixgal force N re-
main constant over each segment • Therefore, they can be removed from the dif-
ferentiation sign# Then, eq.(!L0«4) can be rewritten in the form
M"-)i.m==Q, (10.5)
where M = Ely'' is the bending moment in the blade section and |jl^ = •
EI
The solution of eq«(l0.5) can be written as
M^ = -AsinhtJ'A: + ^sinh|x^, ( 10 #6 )
where the coefficients A and B can be obtained from the boundaiy conditions. / , 145
Thus, for the segment 1 - 2, we have M- = M. at x = and M^ = Mg at x = li^ .
Substituting these conditions into eq.(l0.6), we obtain
^4_ ^2 M^ . \
Here, a^ - W-iHs ^.nd yii = V -•
Mth consideration of eq.(10.7) and of the fact that M^ = Eligy", eq.(l0.6)
can be written in the form
EIy" =
Mn M,
sinhp^ -j- M^coshix^x
.sinKaj tanhaj J ^ ^ ' ^ (10.8)
After twice integrating eq.(10.8) and bearing in mind that at x = y' = Pi;
y = yx, and at x = t^y' = pgi 7 ^ Jsf ^^ obtain
^(i/2-i/i)=^iA^2-f-^i^i + ?i (10.9)
or
^l(^2-i/l)=-^1^2-^l^^l+?2. (10.10)
Here,
M2
154
^12^12 \tanhai J
The equation of deformation for the segment 0-1 can t»e written by analogy
with eq*(10.10):
^0 (i/i - i/o) = - ^0^1 - ^0^0 + f*i- / ^Q -|_^N
Changing all signs in eq.(lO.ll) and adding with eq.(l0.9)^ we obtain
doMo + c^M^^'d,M^ = A,, (10.12)
where
^1 = ^0 — <^i;
After performing the same operations for other adjacent segments, we oId-
tain a system of z equations of the following form:
Table 1. 12
J^o I ^0 L^
;
"o
.^.»
^0
= .
"»
»,
<^z
"l
• ••
• • •
• • •
«• •
• ••
• • •
■ ••
• *•
. . _
<i^.,
^C-2
^^■2
^^.^
-^z-z
'*.r
>",-»
We have written this system of equations here in the form of a table. /1^6
Any of the equations of the system represents the s*um of the products of coeffi-
cients, occupying one row in the rectangular Table 1.12, while the unknown func-
tions Ml simultaneously entering several equations and shown in the vertical
column are given in a separate row on top of Table l.!L2. The unknown function Pq
entering only the first equation is written in this row. The right-hand sides
of the equations A^ are placed in a special column.
The system of equations in Table 1.12 is solved by the method of elimina-
tion of unknowns. This method was already described in Subsection 5 of Section 4»
Thus, the system of equations written out above permits determining the
values of the angle of rotation of the blade at the root Po ^^ ^-H values of
the bending moments M^ if the deformation mode of the blade is known as a set of
values of y^ .
155
To detemdne the elastic force C^^ it is necessary to perform a number of
successive operations, the first of which involves solving the system repre-
sented in Table 1.12» It is expedient to include in this sequence of operations
a determination of the angles of rotation of the elastic blade axis Pj which are
needed later for calculating the aerodynamic forces:
h = ^ iVi - yi-i) - ^i^i^i + ^/-i^/-i.
(10.13)
From the known values of M^ and from the condition of equilibrium of the
elements, we can calculate the shearing force Qi^i+i which is constant over each
segment of the blade. Actually, equating the sum of the moments of all forces
acting on the segment i, i+1 to zero, we obtain the equation
Qi,i^Ai^i=^u+i(yi^i-yi)'\-Mi~Mi^^,
(10-L^)
from which we can determine the value Qi^i+i
M
V
/
"^h
*^
^
Hinge
^ model
4J='2 _
r^
:/h.
\ /
\"n
&^
i\
1
t
\
\
^
t .
/
1
VwV
^
%
'"N
A
I\
.V-
Beam, n
Bt'am li
odel ^
1
iT
V
^4^
\
//
"i ■
^
^.
.y
/'
, Z=72 ,
n
V
\
iw_
1,'
'/
/
\
/
1
1
\
Fig#l#53 Bending Moments with Respect to
the Pirst Overtone of Natural Vibrations,
Calculated with a Different Nimiber of
Masses •
Knowing the value of the
shearing forces over the blade
length, we can determine also
the elastic force Ci appHed to
the mass m^ by the adjacent seg-
ments :
Q— ^Q/,/+i~-Q/-i,;*
(10-15)
These con^^utations permit
deteratlning all values of elas-
tic forces Cj exerted on the
given mass m^ by the adjacent
segments, if the deformation
mode yj is known.
3* Characteristics of Numerical
Integration of Eqs>(jp.l)
In Section 9, we described
the basic characteristics of
application of numerical inte-
gration to the solution of dif-
ferential equations of elastic blade vibrations. It was shown that the success
of numerical integration is largely determined by the magnitude of the Umit
step, which is directly associated with the smallest vibration period of the
mechanical model examined as a blade analog. The limit integration step should
not be too small, since calculation in this case will be extremely time-con-
suming.
A characteristic of the model under study is that it may have as many natu-
156
ral vllDration harmonics as there are segments into which the blade is divided
over its length in the calculation. As already mentioned above, to reduce er-
rors when char]ging from a blade to its mechanical model analog, the blade must
be represented ty at least 25 - 30 segments with the same number of concentrated
masses • Therefore, in determining the limit integration step in this case it is
necessary to proceed from the period of the highest (30"*^^) overtone of natural
vibrations .
Figure 1.40 shows the relation of the natural vibration frequency and /1Z^7
period of an ordinary helicopter blade as a function of the number of the over-
tone. It follows from this diagram that the period of the 30"*^^ overtone of natu-
ral vibrations is about 1*^ with respect to the rotor azimuth. It was stated
above that, in using the most suitable method of numerical integration to obtain
a nondivergent solution, the integration step should be less than one third of
the period of the highest overtone. Consequently, for the method of calculation
examined here, the integration step should be at least 0»3^ with respect to the
rotor azimuth. This stabilizes the solution and permits neglecting the appreci-:
able- error in deterTiiining the airplitudes corresponding to high vibration over-
tones, since their magnitudes are usually small and stresses in the blade are de-
termined mainly by several first harmonics of natioral vibrations . The vibration
anplitude with respect to these harmonics can be determined with satisfactory
accuracy.
It becomes understandable from the above considerations that, in using the
calculation method with a direct determination of the path of motion of points
of a blade, it is advantageous to use a model with a minimum number of concen-
trated loads. It is desirable to use only models with a number of loads not more
than 12 - 15- It should be noted that, with such a small number of segments,
the above beam model introduces errors into the calculation associated with
specific features of this model. To illustrate this. Fig. 1. 53 shows the mode of
the bending moment corresponding to the first overtone of natural blade vibra-
tions, calculated for z = 2S (solid line) and z = l2 (dashed line). It follows
from Fig. 1.53 that, for a small number of segments, the bending moment in the
blade model begins to show peculiarities characteristic for highly flexible beams
stressed by transverse forces in a field of centrifugal forces in that bending
moment concentrations appear at the site where the masses are located. This
characteristic was mentioned already in Section k, Subsection 9* The occurrence
of such concentrations substantially reduces the calculation accuracy. There-
fore, the use of beam models with a number of segments less than 25 (2 = 25) is
not recommended. For a small number of masses, such errors do not arise when /2U8
using a multihinge articulated model, although vibration modes of higher harmon-
ics will be severely distorted. In Fig.l.53j the bending moment calculated for
a multihinge model with a number of segments z = 12 is shown by a dot-dash line.
Proceeding from these considerations, let us examine in greater detail the
method presented here as related to a multihinge model. Furthermore, it will be
shown in Subsection 6 of this Section that a multihinge model permits applying
the calculation of elastic vibrations l:iy numerical integration methods, at an in-
verse order of determining the variables, which is practically iiqDOssible in the
beam model.
157
4- Ecfuations of Motion for a Multihinge Articulated
Blade Model
Let us represent the "blade as a chain consisting of perfectly rigid weight-
less links interconnected iDy hinges. The weight of the blade is concentrated in
the hinges of this chain in the form of individual loads with a mass m^ . The
flexural rigidity of the blade is also concentrated in the hinges, "based on the
concept that a spring of rigidity c^ preventing fracture of the blade in this
hinge is, so to speak, built into each hinge (Fig.l.54)»
Fig. 1,54 Diagram of Multihinge Articulated
Blade Model.
The system of differential equations of vibrations pertaining to this blade
model will be derived here, starting with the equation describing the equilibrium
of the load with the oixiinal number i = 2# Then, by analogy, we will construct
all remaining equations of the system.
The equation of equilibrium of the load with mass m^ can be written in /149
the form
^2i/2=^^2 + ^2-
(10.16)
The elastic force Cg exerted on the mass mg by the adjacent segments of the
model is determined by the formula
158
^2 — ^23 ^12*
(10.17)
where Q^g and Q33 are the shearing forces on segments of the model adjacent to
the load#
To determine the magnitude of the shearing forces Q12 and Q23, we will de-
rive equations that equate to zero the sviti of the moments of all forces relative
to the point of the load with a mass mg (point A) for both segments of the model
adjacent to this load. These equations have the following form:
Qi2A2-^i2(^2-^i)Hr^2-^i = 0;
Q23^23 — ^23 (^3 ~ I/2) + ^3 — ^/2 = 0-
(10. IB)
Determining Q12 and Q33 from this and substituting these values into for-
mula (10.17), we obtain
C2 — Q23 — Qi2 —
M2 \ M2 ^23/ ^23
M2 \ ^12 '23 / *23
■M,
(10.19)
The bending moment entering this equation can be expressed by blade ele-
ment displacement, using the formulas
A^l-^i(Pl2"?0l)
M2
^1
/o-^if^ + V-l^i+T-^2;
i'o-
/oi ^^ "" V ^01 ^12/
= ^2(?23-?12)^'^ //l-^2(-?- + -r)i.'2-y-l/3;
In V hi ^23/ ^23
M-
hz \ hz ^34/ ^34
i/4-
(10.20)
Substituting eqs.( 10.20) for M^, Mg, and M3 into eq.(l0.19), we obtain the
following equation:
C2 = ^li-'o + ^lyi -r /2i;'2 + ^2yz + ^3l/4.
(10-21)
where
^01^12
d,=
^53^;
23^34
e,=-
^1 ^12 / ' ^12 ^12 Vl2 ' ^23 / *
(10.22)
159
IHIIIIIIIIIIII
hs Vl2 ' ^23 / hz ' hz V23 ^34 /
7 2 — " f .2 *'2 I; ' / J /2
M2 ^12 ^M2 ^23 ^ *23
A^23
^23
If we also -write out all remaining values of Cj and substitute them into /150
eq.(l0.l6), then the system of differential equations of blade vibrations can be
represented in the form shovm in Table 1.13.
Table 1. 13
yc
yi
Vi
h
y,
h
...
^z-i
H
fo
h
d,
So
fi
i,
H
d.1
e>
ft
h
'i-i
di
ez
h.
^i
in
• • I
...
...
• « *
...
...
...
* • •
• • •
f • •
• • •
...
• • •
ft*
• • •
•i-x-l
«2-2
fx-1
«!-;
(iz-1
ex-1
/::
^oyo
^i'yi-h
miyfi-h
'"jyj^T-j
^t'l'yz-r'^z'i
^xyz-T^
E^ch equation of the obtained system, occupying one row in Table 1.13, rep-
resents the siom of the products of the known coefficients di, ei, and fj and the
variables Ji which simultaneously enter several equations. The variables yi are
set off vertically in a special row in the upper portion of Table 1.13. The
right-hand sides of the equations, representing the simi of inertia and aerody-
namic forces, are given in a separate column to the right of Table 1.13-
This system of equations directly correlates blade deformations with the
forces acting on the blade, without intermediate coupling across bending moments,
as had been the case in analogous equations pertaining to the beam model de-
scribed above in Subsection 3 of this Section and in equations used previously
in Section 4 for calculating the free vibrations of a blade.
This form of differential equations greatly siirplifies the calculations in
determining the elastic blade deformations, but it also has certain shortcomings.
One of these, as already mentioned, is that the elastic blade axis is not repre-
sented as smooth but as a broken line. The mode of the distribution of the bend-
ing moment over the blade length is also represented as a broken line. A second
shortcoming is the arbitrariness in selecting the hinge rigidities Ci •
Let us present one of. the methods of determining these rigidities. For
this purpose, we investigated two adjacent blade segments. The value of the
160
II I III
III III
hinge rigidity Ci is determined from, the stipulation that the angles of rotation
of the ends of adjacent segments Po ^^ Ps of the equivalent l^eam model coincide
with the angles Poi ^^ Pis ^or the hinge scheme Fig« 1#55):
(?2-Po)^.a«=(Pl2-Pol)>^/.
fffe'
(10.23)
If, in coirparing these angles, we neglect the effect of centrifugal forces
and assume that the "bending moment over these two segments is constant (Mq = /I5I
= Ml = Mg = const), then the condi-
tion (10.23) will yield the follow-
ing formula for determining the hinge
rigidity:
^ben^consi
_L -i^ j_ Jii.
BL
01
BIv.
(10.24)
^Atam
In practice, these assuirptions
can be ol^eyed only approximately.
This leads to certain errors in using
such a calculation method.
5« Sequence of Operations in Calcu-
lating Elastic Vibrations by the
Numerical Integration Method
As a whole, "blade calculations
by the proposed method are carried
out in the following sequence: At
the initial time, which is usually-
related with the azimuth ilf = 0, an
arbitrary blade deformation mode y^
and the distribution of the rate of
displacement of the masses yi are
prescribed. If all values of ji are
known, the elastic forces C^ can be determined by the fonnulas presented in Sub-
sections 2 and 4 of this Section. The angles of rotation of the elastic blade
axis Pi should be calculated at the same time. For the beam model, these are
derived from eq,(10.l3). For the articulated model, they can be determined as
the half-sum of the angles of rotation of two links of the model adjacent to the
point in question:
Fig. 1.55
For Determination of Hinge
Rigidity.
h
2
If the values of p4 and ji are known, then the aerodynamic forces Ti can be
obtained from eqs.(9»205. These data suffice to determine the values Ji by
means of eqs.(l0#l).
161
Next, a change-over is made to the next azimuth of the "blade "by means of /152
fonnulas analogous to eq.(9»4-3):
^(CU.,+7^U^.);
//+A/
ylU='yt + ^^yt+'^^i'ya^;
y\\^i=yt+^^yayi
(10-25)
The values of y^^At* ^t+At* ^^ 7?+ At ^^^ "t^® "^i^® t + At are considered
final. The index i, referring to the number of the concentrated load, is omit-
ted in eqs#(10«25) so as to prevent excessive conpHcation of the e:3qDressions .
All operations are then repeated, to change to the new azimuth. This is
continued for several revolutions of the rotor until the motion of the blade "be-
comes stable. The calculation terminates with the revolution at which the solu-
tion converges to the established solution, with the prescribed accuracy. The
accuracy of the solution is determined by the difference in the ordinates of the
mass displacement when calculatiiig the motion in two successive rotor revolu-
tions.
Analysis of the results can be carried out in any manner, depending on the
purpose of the calculation. To solve problems of blade strength analysis, the
bending moments Mj, every 10° of rotor azimuth, are usually read into the ex-
ternal memory. After calculating the values of M^ and the drag moments of the
blade sections, the values of the stresses and their anpHtude are determined
^may , —
aA=z ^^^i
min J
(10.26)
and the stresses are expanded in harmonics.
Calculation of elastic vibrations by the above method conprises a constant
repetition of the same operations, which amounts to a determination of the
forces Cj and Tj and to the solution of eq.(lO.l). Therefore, the coiiputer time
prmarily depends on the number of such repetitions. This number is determined
by only two factors. The first is the duration of the period of changing to a
stable process, which depends only on the correspondence of the initial condi-
tions of steady motion and on the physical properties of the rotor and is inde-
pendent of the method of calculation. The second, already mentioned above, is
the required integration step.
162
6 • Method of Calculation -with Inverse Order of Determirdng
Varialjles in Nijmgrical Integration
In Section 9 and in this Section, we discussed direct methods of numerical
integration of differential equations for the case in which, on changing to a /153
new time, we determined the variable y and its first derivative j, and then the
second derivative j from the differen-
tial equation. Here, we will examine
a method of calculation proposed tj
V»E.Baskin in which these quantities
are determined in the opposite order •
In sequence, we will investigate
three times: the time tj^ at which the
"blade deformation must t)e determined,
and the two times t^^-i = tj^ - At and
tjj_2 = tj^ - 2At preceding this.
Assimiing that the second deriva-
tive y remains constant over each in-
tegration interval, as shown in
Fig. 1.56c, the value of j^^i can be
e^cpressed by jn~B ^^^ Yn-i -
yn-i
yn—i ~ yn—2
At
(10.27)
Fig. 1.56 Change of Variable y and
its Time Derivatives, in Numerical
Integration.
yn~2
If we now assume that the first
derivative y also remains constant over
the integration interval, as shown in
Fig. 1.56b by a broken line, then the
values of j^-i and j^s can be deter-
mined from the formulas
At
yn^\ ~yn~2
At
(10.28)
Substituting eq.(10.2B) into eq.(10.27) yields the expression for J^~x*
yn-i=—^iyn—^yn-i+yn-2)'
(10.29)
ately
If the integration step is sufficiently small, then we can put approxim-
yn = yn-i
(10.30)
163
and write eq.(l0#29) in the form
yn='^^{yn-'^yn-^'\-yn-2)'
(10.31)
SulDstituting the values of y^ into the system of differential equations re-
presented "by Table 1.13, a system of alge"braic equations relative to the un-
knowns j^ is olDtained. As above, this system is written in the form of
Table 1.14 .
In the variables y in Table l.l!f, the index denoting the instant of time is
given as siperscript, while the index referring to the number of the concen- /I5h
trated load of the model - as before - is given as subscript.
In compiliiTg Table 1#14 it was also assumed that the aerodynamic forces cal-
culated for the time t^^i can be set approximately equal to these forces for the
time t„.
Table 1. 14
y/7
y,"
y?
y;
y".
y;
• ■ •
yl
^Mf:
Ho
d-, '
eo
^7
^.
:
d-1
«!
ez
d-s
dj,
e?
ff^
'3
d.
1
• • ■
• « •
• • •
• • •
* • •
!•••
« • •
• • •
* * •
• • •
• • •
• • •
• • •
• • •
• • •
^^ At"-
d,-,
%[yr-^yM-Tr\
Myo'''-'yo'""']
IjyjnlLzyinfjjn-n
^[yJn.2,,2yJn-0J.j<n-»
^Jy'^M-frt"
The assunption (10. 30) permits e^^ressing the acceleration y^ at the time t^
in terms of the deformations y^-g , Jn-i* ^^^ ^n • After determining the inertia
forces as the products formed by the masses mi with the corresponding accelera-
tions and after adding these to the aerodynamic forces, we can obtain the total
external forces acting on the blade. Then the deformations y^ are detennined as
in a conventional static problem. This is based on solving the system of equa-
tions in Table 1.14^ The only special feature of these equations is the fact
that the conponents of the inertia forces e^^jressed in terms of the still uncal-
culated values of j^ are transposed to the left-hand side and are determined
simultaneously with solving the system of equations*
Thus, the determination of the various parameters of blade motion by this
164
method is carried out in an unconventional order • As it were, first the accel-
erations and then the defonnations are determined. For this reason, we called
this method of solution "inverse method of numerical integration". Another fre-
quently used designation is "inplicit method".
The calculation with the inverse method of numerical integration does not
result in a divergent solution, even at rather large integration steps* There-
fore, the size of the required integration
step should be determined only on the basis
of the magnitude of errors resulting from
the use of this method. The magnitude of
the error can be estimated by applying the
inverse method of integration to the solu-
tion of eq.(9.24)« The results of this cal-
culation are plotted in Fig. 1. 57.
It follows from these calculations that,
to achieve a satisfactory accuracy of the
deformation values corresponding to frequen-
cies equal to the rotor rpm, the integration
step should be less than 1° with respect to
the rotor azimuth /"aT = ^\ .
V 360 ;
Fig. 1.57 Results of Numerical
Solution of Eq.(9.26) by the
"Inverse Method of Integration",
as a Function of the Relative
Integration Step •
In calculating by the method with in- /155
verse order of determining the variables,
the system of equations in Table I.I4 is
solved in sequence at each azimuth, using predetermined values of yl~^ and yi""*" .
At the initial time, these quantities can be taken arbitrarily.
The above method of calculation is more laborious than methods that use ex-
pansion of the solution in accordance with prescribed vibration modes, and thus
is very time-consuming in calculations on digital conputers. However, the method
offers valuable advantages in estimating the influence of various concentrated
effects on a blade, for exanple, in estimating the effects produced by blade
dampers and in all cases when the solution cannot be represented with sufficient
accioracy by a limited number of prescribed modes •
7- Comr>arative Eyaluation of Yarioug Methods of Calculating;
Flexural Blade Vibrations
In this Chapter, we have presented a large number of methods for calculating
flexural blade vibrations; naturaUy, this raises the question as to what method
to select for practical application and what criteria to use as basis for this
selection. The answer is quite sinple: For practical pioxposes, the optimum
method will always be the one that most fully and acc\Jirate3y takes into account
all characteristics of rotor behavior, including the variable induced velocity
field and the nonlinear character of the dependence of aerodsmamic coefficients
on the angle of attack and the Mach number. However, it is inpossible here to
disregard the existing limitations that the more conplete and more accurate the
method of calculation, the more time will be required for calculation on digital
165
H
TABLE 1.15
Differential
equations
Form of presentation
of solution
Method of Calculation with Expansion of Solution
in Eigenfunctions and Determination of the
Coefficients of Expansion of Time Factors in
Fourier Series in Harmonics
Method of Calculation with
Expansion of the Solution in
Eigenfunctions and with
Determination of Time Factors
by the Method of Numerical
Integration of Transformed
Equations
[Ely"]" -[Ny] tmy=T
'Kco'CiCOs^-dfSWip-C2Cos2{ff-d^sin2(p...}'y^^^ t
•i-ieo - Bf cos ^-fiSLT) ij; -e2C0s2\li-J2Sin2tp...}* y^2) -h
*l9o- SiCOS\^-hf swill - giC0Sl\i)-h2Sin2ip^,.]-y^^^
J
Method of Calculation with
Direct Determination of the
Trajectories of Motion of
Individual Points of the Blade
^ik = Ci 7i
yi-f(t)
Method of transforma-
tion of equations
Method of B. G.Galerkin
Equations are not transformed
Method of determining
time factors
Coefficients of expansion of time factors- are
determined from a system of algebraic equations
.r
Direct method of numerical
integration
Inverse method of numerical
integration
-^ S ^ S -^
V ^ K ir> m
g (d to- m
, C V G
O -H *J ^ C
« * .H S
u ^ w » o
*J a-H
« r c
•H O -H
fcr o e
V U
*^ -a V
U V
A = const
'req
5000 <^erations/sec
^req " 20000 operations/sec
^req =100000 operations/sec
Ttie method is convenient for taking into account
a nonuniform induced velocity field expanded in
harmonic components
The methods are convenient for taking into account induced
velocities determined for a rotor with finite number of blades
(for a number of points over the radius Zr -12^
req
50000 operations/sec
req
100000 operations/sec
req
= 500000 operations/sec
'-as
A = const
Cx=f(^,M)
A-var
For taking into account the relation Cy = f(a,M)
and Cx = f(a,M), this method is unsuitable in
practice
The methods are convenient for taking into account the nonlinear-
relation Cy = f(a,M) and q^.= f(ajM)'
^req =" 500000 operations/sec j V^eq = 200000 operations/sec
at 2r-t2
Vreq = 250000 operations/sec I V^.^^ > 100000 operations/sec
conputers. Therefore, in selecting the optimum calculation method, the main
criterion is the machine capability which places a limit on the use of the most
refined calculation methods •
To select the most suitable calculation method, we conpiled Table 1.15 which
also gives the required speed of computation for various methods of calculation.
The table also shows the basic characteristics of the different methods.
Here, we will present approximate values of the required speed of operation
per second Vp^q for carrying out a calculation within 5-10 min. The required
speed is given for all calculation methods in four variants of the assunptions
used. The required capacity of the conputer memory is not estimated in Table 1.15
since, in modern conputers, this usually constitutes no handicap for the pro-
grammer .
It follows from a perusal of the data in Table I.I5 that, for a low-speed /157
cooputer (speed of the order of 5OOO operations/sec), only one method known as
the method of calculation in harmonics can be used to any greater extent. In
this method, the solution is expanded in eigenf unctions . The time factors in
these functions are represented as a Fourier series in harmonics. The coeffi-
cients of this series are determined from a system of algebraic equations derived
from a differential equation by means of Galerkin^s method. This method is de-
scribed in Section 8.
On low- speed conputers, this method can be used only under the assionption
of a unifomi induced velocity distribution X = const. For taking into account
the nonlinear dependence of aerodynamic coefficients on the profile angle of at-
tack and on the Mach n-umber, the method is practically useless. These relations
can be considered in this method only by making extensive assunptions. However,
even with such an approach, the computation necessary for constructing the
mathematical formulas is so laborious that, for all practical purposes, it "is
siirply unfeasible.
On moderate-speed conputers (speed of 20,000 - 50,000 operations/sec), the
most convenient method is to expand the solution in eigenfunctions and to deter-
mine the time factors of these functions by numerical integration. This method
is presented in Section 9, and is quite convenient for taking into account the
norOinear correlation between aerodynamic coefficients, profile angle of attack,
and Mach number.
In cases in which the variable induced velocity field must be considered,
this method can be used only on high-speed conputers. If, in determining the in-
duced velocities, the number of calculated points over the radius and azimuth of
the rotor is limited, the calculation can be performed also on moderate-speed
conputers .
The method of calculation with direct determination of the trajectories of
motion of individual blade points (see Sect.lO) can be used only on conputers
with a speed greater than V > 100,000 operations/sec. A consideration of the
variable induced velocity fields and of the nonlinear correlations between aero-
dynamic coefficients, profile angle of attack, and Mach number further increases
the speed needed for this method. Only the inverse numerical integration method
167
is considered in the last coltunn of TalDle 1,15 . In using the direct method of
numerical integration, the required coHputer speed for the method with direct de-
termination of the trajectories of motion of individual tilade points may in-
crease even more steeply.
The required computer speeds given in Table 1.15 are obtained for the case
in which the confutations require 5 " 10 min. Within this time, it is possible
to make several checkouts required in designing a blade with variation in rotor
parameters and flight regime.
If we limit ourselves to a calculation at only one variant of the parameters,
a longer conputer time is admissible. In this case, the required coirputer speed
shown in Table 1.15 can be reduced accordingly.
Using the above considerations, it is possible - for each individual case -
to select the optimum method based on the possibility of using various assunp-
tions and available time in calculations on a computer.
Section 11. Fatigue Strength and Blade Life /I5g
1. Testing a Structure to Determine its Service life
The service life of a given struct'ure is usually established on the basis
of results of dynamic analysis.
Depending upon how essential the structure is for flight safety, tests of
one or several design variants are performed. Frequently, only individual parts
of a structure whose strength is decisive for the entire unit as a whole, are
tested.
In determining the blade life, it is conventional to test individual spar
segments with airframe components that set up stress concentrations in the spar.
Specimens of at least three different spar segments are tested. As a rule, these
segments include the root butt and two segments along the length of the spar.
Sometimes it becomes necessary to test additional specimens to check individual
design features of the spar (for example, at points of transition of the spar
cross section) .
Specimens of blades are almost always tested on resonance stands, with ex-
citation by mechanical vibrators. The length of the test specimen is chosen
such that its natural frequency in bending is within the operating range of the
vibrator. Usually the tests are conducted at a frequency of 1500 -to 2500 cycles
per minute. In this case, the length of the specimens is of the order of 3-4 ni.
In addition to alternating bending stresses, the specimen must be extended by
longitudinal forces creating a constant static load close to that which the
blade esq^eriences in flight due to the effect of centrifugal forces. Figure 1.58
shows a stand for testing of helicopter blade specimens x^th a centrifugal force
of the order of 100 tons (force).
In general, conplete blades rather than individual short specimens are
tested, because of the excessive conplexity of test stands required for- this pur-
168
pose and the long testing time, since the vi'bration frequency in this case can-
not be higher than 300 - 400 cpm.
2- Dispersion of the Characteristics of Endiorance
in Fatigue Tests
An appreciable scattering of the test results is observed in fatigue tests
on a certain number of specimens manufactured under identical conditions. Fail-
ure of specimens tested at the
same stress level occurs at a /159
different number of cycles N.
The ratio of the greatest nimiber
of cycles to the least number
often reaches 20-40.
The dispersion of the char-
acteristics of fatigue strength
is explained by the inhomogeneity
of the structure of the material
and by the difference in the con-
ditions of manufacturing and pro-
cessing the specimens. Failure
of specimens always begins from
small flaws within the material
and on the surface' of the speci-
men. In the overwhelming ma-
jority of cases, failure begins
from a defect situated on the
siorface. In this case, the en-
diirance characteristics of the specimen are determined by the type and magnitude
of these defects.
The scattering of fatigue failure data in tests of various specimens is
usually characterized by a distribution function of the ntunber of cycles N to
failure of the specimen. An analysis of the test data indicates that the dis-
tribution of logarithms of the ntonber of cycles log N to failure rather closely
obeys the normal distribution law at almost all average values of the probabil-
ity of failure, beginning approximately from a probability of O.Ol - 0.02.
Figure 1.59 shows the distribution of the probability of failure P and the
probability density cp, corresponding to the real characteristics of endurance of
the structure (solid curves) and those determined by the normal distribution law
(broken curves):
Fig. 1.58 Blade Test Stand.
{logN-mi^l^)2
<9{1<^N)^
2S\
SlcjN "/Sjt
^-e
logN
(11.1)
P(ioffN)= ^<f(.i)dt
(11.2)
169
Here,
cp(log M) = probability density function of failure of the structure;
P(log N) = probability of failtire of the structure at a number of cycles
of stress less than N;
5 = log N = value of the logarithm of the number of cycles to failure of
the structure;
Slog N ~ niean-square deviation of the distribution of the logarithms
of the number of cycles to failure of the structiore;
miog 1^ = mathematical expectation of the distribution of the logarithms
of the number of cycles.
In the range of low probability of failure, the distribution function usu-
ally deviates from the normal law (Plg.1.59)* This has to do with an important
feature of the characteristics of endurance • In fact, fatigue failure can take
place only after a certain number
of cycles of stress Nq and can
never occur earlier • This feature
m ^^ of the characteristics of endior-
/ ' \ ^^sf^^^iuqti) ance leads to the concept of a
I \\^^^^ zone of insensitivity to N in which
)ff\ "the probability of failure of the
y^y^ ' \ structiire is equal to zero (P = O).
• ^.-^ X / 1 \ -^^ particular, this penaits an
^^ l/ / j \^^^{iogn) inportant conclusion as to the pos-
^^ \\ y/ j X^. sibility of determining the service
^"^ ik^ L ^^'-'-T— ■■ ;^ life of the structure, based on
logn^ frii^N logM endurance conditions with a prob-
ability of failure equal to zero.
Fig. 1.59 Curves for the Distribution of even in the presence of sufficient-
Endurance in Tests and Corresponding to ly high alternating stresses.
Normal Law.
Unfortunately, a determination
of the sensitivity threshold Nq,
with any satisfactory accuracy, is virtually inpossible. Therefore, in deter-
mining the service life of a given structure the distribution law of endtirance
is usually taken as noniial,and the requirement P = is replaced by the require-
ment of a very low probability of failure.
/160
The deviation of the values of the logarithms of the number of cycles from
the normal law should be observed also in the region of high probabilities of
failure. At a relatively low level of alternating stresses this happens since
there is almost always some specimen that does not fail even at a very large numr-
ber of cycles of stress.
3. Basic Characteristics, of the Fatigue Stren^jbh of Structure
The fatigue strength of a structure is characterized usually ^yj the number
of cycles N it is able to withstand prior to failure at a given upper limit of
alternating stresses a. The higher the ipper limit of alternating stresses a,
the smaller the number of cycles of stress the structure will resist.
170
1
The curve characterizing the number of cycles N to failure as a function of
the -upper limit of alternating stresses a is called the Wohler curve.
The Wohler curve can "be approximately described ty the equation
a=o^=:=const at A;^>A^^. / (11-3)
Here,
Oyt = maximum anpHtude of stress "below which the structure will with-
stand an indefinitely large numlDer of cycles of stress N without ^
fractiore; this peak ajrplitude is usually called the fatigue limit
or endurance limit;
N„ = minimum n-umber of cycles of stress corresponding to the fatigue
limit;
m = some exponent whose value is determined from test, results.
The Wohler curve can he plotted for different values of the probability of
failure. For this, a batch of test pieces is divided into several groijps and
tested at different ranges of alternating stresses.
After constructing the distribution functions of endurance for various
levels of alternating stresses (i^g.l.60) and connecting points of the same prob-
ability of failure, we can obtain the Wohler curves corresponding to a different
probability of failure. Usually, in so doing the dispersion of the characteris-
tics of endurance is smaller, the higher the level of alternating stresses, and
the sensitivity threshold Nq is more distinctly expressed at lower stresses. At
low stresses, the sensitivity threshold, is reached at relatively high probabili-
ties P, whereas at high stresses it shifts toward such small probabilities that
it usually goes unnoted. /161
The test data almost always confirm the presence of a fatigue limit a„ . At
a given stress a a certain number of specimens usually will not fail even at a
very large number of cycles of stress. The existence of a fatigue limit is con-
firmed also in practical experience of operating various machines and mechanisms.
We know of many different conponents that constantly operate under appreciable
alternating stresses and do not fail at a number of cycles of 10^ and more.
There are individual exceptions to this general rule. It has been noted that,
for certain structural elements made of al-uminum alloys, the fatigue curve con-
tinues to drop even at an endurance of the order of 10^^^- lO"*"*^ cycles. However,
this drop is so negligible that even in this case the Wohler curve can be ap-
proximately represented in the form of eq.(ll.3). In any case, as applied to
the basic parts of a helicopter, consideration of this drop yields no substantial
refinements.
A definite dispersion is also observed in the values of the endurance limits.
The presence of a sensitivity threshold with respect to the anplitude of stresses
is characteristic for their distribution. This threshold will henceforth be
called the minimum fatigue limit Owmin' ^'^ stresses below Owmii^ not a single
specimen will fail, even at a very large number of cycles of stress. In con-
171
g kg/mm^
Fig#1.60 Distribution of Endiorance at Different Levels
of Alternating Stresses.
<5 kg/mm^
10G
SO
30
10
10
mz
30
K
■^?
P-50%
10
vv
^^^.
,P-5%
"-^
^
=»•— -—
£j;s_
m C 1
!
10
ir
P-0
1
I
1-10'
^'10'
6-10'
8'W^
v////////w///^^
V////////J^^^^^
logN
Fig. 1,61 Wohler Curves Correspond-
ing to Different Probability of
Failure.
Pig. 1.62 Wohler Curves Corresponding
to Different Probability of Failure
on a Logarithmic Scale.
f ormity with the above features of the characteristics of fatigue, the l//6hler
curves should have the slope depicted in RLg.l#6l.
If the curves corresponding to a different probability of failure are re-
placed by an approxmate analytic relation (11.3), then the W6h3^er curves on a
logarithmic scale vail have the slope shown in RLg.l.62» The zone of insensi-
tivity corresponding to zero probability of failure is hatched in this graph.
172
f
At such a plotting of the l//6hler curves, the nuinber of cycles N„ corresponding
to the endurance limit and the e:xponent m differ for curves corresponding to
different proT^ahilities of failure.
It should he noted that constzniction of the Wohler ciorves as shown in
Figs.l#6l and 1«62 is possible only in tests with small laboratory specimens,
since a very large number of test pieces is required.
Construction of such curves is practically inpossible when estimating the
strength of a structiire, since only a very small number of specimens can be used
in such estimates. Often this n-umber does not exceed n = 3 - 5 (where n is the
n-umber of tested specimens). In this case,
the tests yield only n values of the num-
ber of cycles to fracture at a prescribed
magnitude of loads for a strength esti-
mate. With such a Umited number of data,
some idea on the fatigue characteristics
of a given structure can be gained only on
the basis of certain assumptions with re-
spect to the Wbhler curves.
Conp
ression
80 <3mkg/mm^
Fig. 1.63 Hay's Diagram for Speci-
mens of Tubular Blade Spars.
The range of alternating stresses at
which the structure will withstand a pre-
scribed number of cycles N to failure de-
pends also on the magnitude of the con-
stant conponent of the stresses of the
cycle a„ (static load). The greater the /163
static load, the smaller the range of
stresses at which the structure will with-
This dependence is usually characterized by
stand a given number of cycles
Hay's diagram. As an exairple Pig. 1. 63 shows the configuration of such a diagram.
For tubular steel spars at a „ = 20 - 30 kg/mm^ an increase in static load
by an amount Aa^ leads to a decrease in the fatigue limit by an amount Aa„ c^
^ O.^AOn . For diaralimxLn spars at a^i =6-8 kg/mm^, the value is Aa„ f:^ 0.3Aa5, .
It should be mentioned that, in the region of constant compressive stresses,
the fatigue Umits increase substantially. This fact is utilized when confer-
ring strength to conponents by cold-working (see Subsects.l6 and 1?).
4* Stres_s_es Set Up in the Blade Structure in FHght
In Section 1 of this Chapter (Subsect.3) it was mentioned that, under the
effect of aerodynamic forces, the blades of a helicopter in flight are subject
to appreciable alternating stresses in two different types of regimes designated
as low-and high-speed modes.
Figure 1.64 shows the type of variation in ajrplitudes of alternating
stresses with respect to flying speed, for two blade structures: one with a steel
and one with a duralumin spar. As indicated in this diagram, maximum alternating
stresses can arise both at low speeds (braking regime) and at maxLmim flying
173
speed. As demonstrated "before, the "blades perform flexural vi"brations such that,
at each point of the spar, the stresses vary in accordance with a periodic law
duplicating each revolution of the rotor. As a typical exanple. Pig. 1.65 shows
the recoi*ding of stresses o^btained in iDlade sections at relative radii r = 0.73
and r = 0.8 in horizontal flight at rela-
tively high speed. The same diagram gives
the harmonic content of the stresses set vsp
in these blade sections.
Usually, in a horizontal flight at jj, =
= 0.2-0.4 the first harmonic conponent of
the stresses reaches maximum values. The
second harmonic is lower in anpHtude and
generally amoimts to 30 - 70^ of the first
harmonic. The first and second harmonics,
generally totaling 70 - 90^, determine the
magnitude of the total alternating "blade /2£U
stresses in these regimes, since the higher
harmonics usually are small. Their magni-
tude almost always decreases with an increase
in order of the harmonic. Such a type of
variation in magnitude of harmonics can be
attributed to a decrease in magnitude of the
harmonic coirponents of aerodynamic forces
on change-over to higher harmonics.
Fig. 1.64 Character of Vari-
ation in Amplitudes of Alter-
nating Stresses as a Function
of Flying Speed in Blades of
Low (Tubular Steel) and Mod-
erate (Duralumin Spar) Rigid-
ity in the Flapping Plane.
For all blades there are exceptions to
this rule, having to do with the occurrence
of or proximity to resonance.
In low-speed modes, the harmonic content of the effective stresses is dif-
ferent. Here the higher harmonics predominate, and harmonics close in frequen-
cies to the frequency of the natural vibrations of the second and third over-
tones are mainly distinguished. An especially pronounced increase in alternating
stresses in these flight regimes (see Fig. 1.64) takes place for blades of low
rigidity in the flapping plane (see Sect .3, Subsect.3). For such blades, stresses
with the fourth and sixth harmonics are predominant (Fig. 1.66). Low-speed modes
may cause damage to the structure of such blades (see Table 1.2l).
For a blade of moderate rigidity in the flapping plane, the increase in al-
ternating stresses at low speeds is appreciably weaker (see Fig. 1.64) and pre-
dominance of higher harmonics is not so marked (FLg.l.67). For such blades (just
as for blades of high rigidity) high-speed flight modes may lead to basic damage
potential.
Along with alternating stresses due to f lexural vibrations, the blade spar
is extended and bent by constant (in magnitude) centrifi:igal forces and by the
constant coirponent of the aerodynamic forces. Therefore, the spar material
works under alternating stresses with a large static load. The static load mark-
edly lowers the fatigue strength of the spar.
174
5* ftypothesis of linear Summation of Damage Potential
and_ Average Equival ent Ang jlitude of Alternating
Stresses
In different flight regimes, alternating stresses of widely differing mag-
nitude are set ip in a structure • In this case, the duration of individual
fUght regimes inay differ substantially. Thus, a flight at crioising speed is
usually the regime of longest duration. In heUcopters used for cargo transport.
Recording of stresses:
revolu tion
of rotor
Harmonic content
7 l^umber
of harmonic
7 Number
of harmonic
s kg/mm
30
/Total stresses
rStresses wi th respect to
fourth and fifth harmonics
1 2 3 't S 6 7 8 Number of harmonic
Pig. 1.65 Recording of Stresses in
Two Sections of a Helicopter Blade
in Horizontal FUght Regime (|jl =
= 0.3) and their Harmonic Content.
Fig. 1.66 Oscillogram of Alternating
Stresses in a Blade with a Tubular
Steel Spar of Low Rigidity in the
Flapping Plane during Braking, and
their Harmonic Content .
this regime occijpies 60 - 70^ of the service Hfe. The maximum flying speed 7166
of cargo helicopters used in the national economy is rarely reached. Such heli-
copters also spend very little time in low-speed modes which generally are only
transient regimes during takeoff and approach to landing.
However, helicopters can be used for widely differing types of work, where
the duration of individual flight regimes varies. As an exanple. Table 1.21 gives
175
1.08
r^O.ZS
1 2 J ^ 5 6 7 8
Number of harmonic
Fig •1.6? Hannonic Content of
Alternating Stresses in a
Blade of Moderate Rigidity
with a Pressed Duralumin Spar
in Braking Regime.
the values of the relative duration of dif-
ferent regimes a^ common for one of the mili-
tary transport helicopters.
The service life of a given structure
should be deteimned with consideration of
the time base of a helicopter in flight re-
gimes of differing alternating stress level
which contribute to the structure a different
portion of fatigue damage potential. To take
this into account it is convenient to use the
hypothesis of linear sunmation of damage po-
tentials. This hypothesis presipposes the
possibility of summing individual conponents
of damage potential contributed by different
stress levels and stipulates that failure of
a structure takes place as soon as
where
AA^s-^l, 1
(11.4)
Here,
Nj = number of cycles to failixre for a steadily sustained stress level
with airplitude o^;
ANj = number of cycles of stress with aiiplitude oi e^erienced by the
structure in the i-th flight regime.
ANi
The ratio AN^ =
Ni
is usually called the damage potential of a structure
in a regime with an anplitude of stresses Oi , while AN^ is designated as total
damageability .
It has been proved by other authors that, at a certain alternation of stress
regimes, failure of a structure may take place as soon as
AiV2<l.
However, the cases discussed in those papers, for the most part, do not
cover stress conditions of the helicopter parts. Therefore, we can almost al-
ways use eq.(ll.4) in the calculations.
As a consequence of the dispersion of the characteristics of endurance, the
damage potential of individual specimens of a given structure may differ even
for one and the same stress level. Structures with the lowest values of endior-
ance are subject to maximum damageability. Therefore, one can talk of damage
potential as corresponding to a certain probability of failure.
If, in eq.(ll.4), the values of Nj corresponding to an assigned probability
of failure Paa^ are given, then the probability of failure will also be equal to
176
Pasd ^"t AN^ = 1. This makes it possible to obtain the formula for calculating
the safe number of cycles of stress Ng "with the assigned probability of failure
Pa8d determining the service life of structures based on endiarance conditions:
K
23L ' (11.5)
Here, ^^
oi^ - — = relative dtiration of the regime vdth stress a^\
Ng. 1 = niMber of cycles of stress during the life of the struc- /167
ture when determining the relative duration of individual
fUght regimes ot^ (generally speaking, we can take any
arbitrary interval of the service time of a helicopter with
a nimiber of cycles N that need not at all be equal to the
n-umber of cycles of stress during the rated service life
of the structure Ng^ | );
Nj = number of cycles of stress of airpHtude Oj at which the
probability of failure is equal to that assigned (Pasd)*
If the acting stresses are lower than the minimum fatigue limit, then
damageabiHty is not introduced into the structure. In this case, the number of
cycles Nj in eq.(11.5) can be set equal to infinity.
Let us introduce the concept of relative dioration of regimes e which add a
damage potential to the structure:
where B^ is the number of cycles of stress during the life of the structiore
which contributes to damageability.
Thus, during the service life R, the structure is damaged only during a time
equal to eR.
It often proves convenient, for greater clarity of the calculations, to in-
troduce the concept of average equivalent amplitude of alternating stresses.
The average equivalent anplitude of stresses is an anplitude constant in
time and acting during a part of the service life equal to eR, which contributes
damageability to the structure equal to the damageability introduced by anpli-
tudes of alternating stresses differing in magnitude in all flight regimes en-
couatered during the service of a helicopter.
In introducing this concept, it is assumed that at stresses greater than
the fatigue limit, the endurance of the structure can be determined as
177
'^'-''-ftr- (11.6)
Then, substituting eq.(11.6) into eq.(ll»4)^ we olDtain
2 ^^i^^T
• ^\, (11.7)
where summation is performed only for those regimes that raise the damage po-
tential to the structure.
If we introduce one equivalent stress level with an amplitude a^q and with
a number of cycles determined from the stipulation that stresses a^q act conr-
tinuously during a part of the service life eR, i.e., that N^^ = ^^bA » then we
can write
(11-8)
Consequently,
°n
=1?^tS°'''-
(11.9)
The calculation of equivalent stresses for a "blade of a helicopter at the /168
maximally stressed section at a relative radius r - 0.74 is given as an exanple
in SulDsection 12; see also Table 1.2l» The ^ar of this blade is a steel tube
squashed into an ellipse over its entire length, beginning from radius r = 0.3*
The minimum fatigue nimit of the tu*bular blade at this station, based on results
of dynamic tests, can be taken as equal to a^^^ = 13 kg/mm^.
Stress analysis of a blade with a tubular steel spar is usually carried out
in two planes: in the plane of minimum (oy) and in the plane of maxmimi rigidity
(ctx ) • 3ii this case, it may happen that, at some point of the perimeter of the
spar section, the anplitude of alternating stresses reaches a magnitude a^ =
= Jo^ + Oy greater than the airplLtude ay .
However, usually owing to the phase difference of the stresses acting in
these two planes, such a magnitude of alternating stresses is almost never
reached. Therefore, in calculating the service life of a blade we can use the
approx5.mate fonnula:
= ^i^ +
^(/"^J + %^-^.).
The coefficient % can be calculated if a simultaneous recording of stresses
and Oy is available. If there are no data for determining §, then sufficiently
178
reliable resialts can te obtained by assuming 5 = 0.5»
6. Dispersion of t he Ainplitudes of Alte rnating Stresses
in__an_A 3siKned Flight Regime
In measioring alternating stresses in fUghb it has been found that, at an
assigned flight regime, the magnitudes of stresses differ during the flight re-
gime and in different flights. This makes it necessary to introduce the average
equivalent amplitude of alternating stresses in all flight regimes.
To determine this amplitude, we can use special oscillogram decoders which
permit determining the number of airplitudes of stresses n^ located in the range
where o^ and a^-i are the anplitude levels of alternating stresses selected for
the calculation.
Then, the average equivalent ajiplitude of alternating stresses in the re-
gime in question can be determined by a formula analogous to eq.(ll»9):
>.=/is«.'-- '''■^'
Here,
n^^ = relative number of cycles with anplitude Qj^
nu=-
where
n^^ = nijmber of cycles with anplitude aj^ while n^ is the total number
of cycles recorded by the decoder;
e^ = relative number of cycles with stresses greater than the minimum
fatigue limit in the i-th flight regime.
Summation with respect to k is carried out only for those time intervals in
the i-th regime in which the anplitude of stresses aj^ is greater than the mini-
mum fatigue limit a^ .
When determining the average equivalent anplitude during the entire service
life of a helicopter by means of eq.(11.9)^ "the anplitude in each flight regime
should be calculated from eq.(ll.lo), while the relative dioration of the regimes
raising the damage potential of the structure is calculated by the formula
i
To sinplify the decoding, it is general practice to replace determination /169
of the equivalent anplitude by the maximum anplitude in each regime; this raises
179
the relia*bility margin "but leads to some decrease in service life of the struc-
ture.
?• Method of Calculating Service laf e with the U se
of ReHaMlity Coefficients
The problem of determining the service life of a given structure reduces to
finding some safe ntimlDer of cycles of stress in service Ng at which the prolD-
ability of failure of the structure is very small and equal to the assigned
value. If it were possible to test a sufficiently large number of specimens, it
would be easy to find Ng after determining the distribution characteristic of
their service life (Fig. 1*68). Numerous methods of calculation of service life
[see, for exanple (Ref .43)] are based on this approach. However, it is usually
necessary to determine the service life of a structure on the basis of dynamic
tests of a few specimens n of the structure, where it is inpossible to determine
the distribution of service life with the required accuracy. Therefore, the
method of calculation of service life of a structure, based on the introduction
of certain margins of reliability with respect to the number of cycles T|n and
anplitude of alternating stresses Tlo-^has become popular in practical engineering.
n.
93-
90-
801
50 \
10 ■
10-
— h-
— \—
j —
=
-
=
7
-U
T"
-+rTr^
—
—
-4
T—
1
1
t-^^
c
c
1
J
M i
^
:=
=
=
-^-
^lagri
,
"
-
Fqj;;
^f]
■* '^tSgN '
h4 ■
^
E
m
r—
1
~
=
1 — '~^
M
=
^^-
^
^^
^
~" ) ft
ii4-
;
'^
^
--
\
rT
1 i
'.^-4U-
^-^'
f
_
p^ .-.-^
-1 1"
— L
—
--+ 1 1 -
— 1
o /UUl
^-Jf-
--
i ;.
^
-\-p
--fH-+
-
i
—
-
' 1
d
-i-
^-
rrW]
0,1
0M1
0M01
logN^ 6
^09^m
logN
Pig. 1.68 Determination of Safe Number of Cycles,
Based on the Service life Distribution Curve.
To calculate the service life by this method, it is necessary to make a
stress analysis of a structure in various flight regimes, to determine the equi-
valent stresses, and to conduct dynamic tests of one or several specimens of the
structure at stresses of
<^/^rf='n«^.
(11.11)
The margin of reliability Tlj is introduced here to take into accoimt the
possible difference in values of alternating stresses in identical units of dif-
ferent helicopters.
180
After testing the specimens and obtaining the minimuni value of cycles to
failure M^ij^, we determine the safe nvniber of cycles of stress in service, "by
means of the formula
^w ' (11.12)
The margin of reliability TIn is introduced to take into account the dis- /170
persion of the endurance characteristics*
Then, the service life of a structure in hours can be determined from the
formula
where f is the frequency of stressing the blade in service (cycles per minute).
In some cases, the endurance of a structure might depend on the frequency
of stress application. Therefore, if dynamic tests are carried out at a frequency
greater than the frequency of stressing in flight, it will be necessary to in-
troduce an additional margin for the frequency of stressing T|f . This margin is
introduced mainly for components made of duralumin and for tests carried out at
a frequency which is by a factor of 5 - 10 higher than the frequency of stress-
ing in flight. In this case, the value is taken as equal to T|f = 1.5 - 2.0.
When allowing for these factors, the formula for determining the service life
can be written in the form
60/T,^,,^ • (11.13)
If we assume that the distribution of the endurance characteristics obeys
the normal law and that the parameters of this law are known, then, as already
mentioned, the magnitudes of the required reliability margins with respect to
the number of cycles T]n and anplitude of alternating stresses Vfj could be deter-
mined by calculation, after assigrdng a certain, sufficiently small probability
of failure of the structure in service. However, such calculations cannot lay
claim to high accuracy. Therefore, we can use the method of assigning the magni-
tudes of these coefficients on the basis of helicopter operating experience.
On the basis of such experience, the safety factor with respect to the am-
plitude of alternating stresses T|a can be taken as equal to 1.2, whereas the
factor with respect to the number of cycles of stress T|n varies as a function of
the number of tested specimens and the degree of essentiality of the unit for
flight safety.
All units and conponents of a helicopter can be divided into four groips
based on degree of essentiality for flight safety:
Group I - units whose failure leads' to immediate and conplete disnption of
IBl
operat)ilit7 and safety, "with a difficultly detectable incipient fatigue crack.
This group includes blades whose spar is covered and does not permit postflight
inspection, a variety of components of the hub and controls of the main and aux-
iliaiy rotors not accessible to inspection, the rotor shaft, etc.
Group II - units whose failure could lead to inmediate and conplete disrup-
tion of operability of the structure and flight safety, but where early detec-
tion of incipient fatigue cracks is possible. This groi^D includes blades with a
reliably operating system signaling the appearance of cracks, as well as all
other units classified in Group I provided that incipient fatigue cracks can be
detected in preflight inspection.
TABIE 1.16
SAFETY FACTORS WITH RESPECT TO
NUMBER OF CYCIES^'"
Safety Factor Tj;y
Number of Tested
Specimens n
Group
I
Group
11
Group
in
Group
IV
1
12'
6.0
6
2.6
2
8
4.0
4
2.0
3
6
3.0
3
1.5
6
4
2.5
2
1.0
Groijp III - units whose failiire leads
to partial loss of operability and en-
dangers flight safety, but permits forced
landing without damage to the helicopter.
This group includes numerous fuselage
parts and even the reduction gear frame-
work if it is redundant #
Group IV - units whose failure
causes partial loss of operability, al-
lows continuance of flight, does not lead
to rapid failure of other units, and per-
mits detecting riipture in ground inspec-
tion. This group includes numerous ele-
ments of the fuselage, stabilizer of the
helicopter, and of other related struc-
tural elements .
7171
""' The factors TIn given for Group I
of the linits are double the usual
values, since they include also
the factors Tj^ often introduced to
allow for inaccuracy of the hy-
pothesis of linear summation of
damageability .
The more essential the unit, the
greater should be the magnitude of the
safety factor with respect to the number
of cycles. The following values of these
factors are proposed here (Table 1.16).
In practice it is possible to re-
alize safety factors for the number of
cycles required in Groups I and II of
helicopter parts only at a very low fre-
quency of stress alternation in flight. In establishing the service life with
such large safety factors for all basic helicopter units, tests up to a very
large niomber of cycles, much greater than 10 '^ cycles, would be required; this
would take a great deal of time. Therefore, an accelerated method of dynamic
tests with a safety factor for the nimaber of cycles of T|n = 1 or of even less
than unity is gaining in popularity. In this case, the required reliability is
secured by introducing only the safety factor for stresses. To convert the fac-
tor Tltg to the factor for Tig we generally use eq.(11.3) with the exponent m = 6.
TflH-th this approach, the required factor of safety for the anplitude of alternat-
ing stresses differs, depending on the number of tested specimens and is greater
for diiralumin, making it necessary to introduce an additional factor for the dif-
ference of the frequency in tests and in flight.
182
TABLE 1#17 Tfith consideration of the aforesaid, it
is possible to adopt safety factors for the
amplitude of alternating stresses indicated
in Ta"ble 1#17 for Groip I of helicopter parts-
In dynamic tests vath such safety factors
T]cr, the safe nunilDer of cycles is determined
with respect to the minimiim n-umber of cycles
of stress of the specimen to failure Ng =
It should "be borne in mind, however, that
the conduction of tests with such large safety
factors for the airplitude of alternating
stresses is possible only if the character of the stress distribution for dif-
ferent conponents of the structure does not substantially change on increasing
the load. If, ipon increasing the loads, there is a redistribution of stresses
as a consequence of say flare-out of joints, occurrence of mutual displacements
of contacting parts which ordinarily work under loads without such displacements,
or for other similar reasons, application of this test method is not recommended.
Number of Tested
Specimens n
%
(TQAr=l)
Steel
Duralumin
1
1.8
2.0
2
1.7
1.9
3
1.6
1.8
6
1.5
1.7
8. Method of A.F.Selikhov for Calculating the Recruired Safety
Factor with Respect to the Number of Cycles T|n
Z122
As mentioned above in Subsection 2, the endurance of a structure has a sen-
sitivity threshold with respect to the rnonber of cycles Nq so that the distribu-
tion function in the region of
low probabilities of failure de-
viates from the normal law.
Theoretically, one could select
a safety factor for the nimiber
of cycles TIn such that the prob-
ability of failure of the struc-
ture would be equal to zero.
However, as shown elsewhere
(Ref .Zj4), for a sufficiently ac-
curate determination of the sen-
sitivity threshold a large num-
ber of specimens is required so
that it usually is inpossible to
determine its magnitude for a
structure. Therefore, one gener-
ally assumes that the logarithms
of the number of cycles to fail-
ure log N are distributed ac-
cording to the normal law and
the service life of the structiire
is not based on the condition
that the probability of failure is P = but on the condition that this prob-
ability is sufficiently small, say equal to P = 1/10,000. If there actually is
a threshold of .sensitivity present, then the stipulation of such a small prob-
Q
/
\ T
\ (^ {logN
mtn)
w
9gN)
^\
/ //\
\
^^jt^^
V^
6.Z
SM
U
6,8
ZO
7.2
lo^N
Fig. 1.69 Distribution of Minim-urn Endurance
Values for a Different Number of Specimens.
183
a"bility of failure, calculated in accordance with the normal law of distribution,
tends to "be more rigorous than the requirement P = which could be inposed if
the value of N^ were calculable • Therefore, a determination of safety factors
on the basis of a somewhat greater probability of failure, say P = 1/1000 and
even P = 1/100, is entirely permissible.
To determine the required safety factors for the number of cycles we can
use the method proposed by A.F.Sellkhov. This method involves the following:
Assuming that the distribution of the logarithms of the numbers of cycles
to failure of a structure obeys the normal law
<p(/^^A^)=-
^lo^N
/2n
(11.14)
then the distribution of the minimum endurance values of a certain batch of speci-
mens of this structure can be determined from the formula
?min (^^^A^) — ^ [l + O (^^
~~h(^N
SlogN V2
/7-1
<o{logN),
(11.15)
where
n = number of tested specimens;
f (x) = Laplace function (^x = °^io»n - log N ^ ^
The character of the distribution of 9^1^ (log N) for values SiogN = 0*15
and n = 5, 10, and 100 is indicated in Fig .1.69 . The values of the mathematical
expectations and the mean-square deviations of this distribution as a function
of SjogN 3,nd n can be foiind from the curves presented in Figs. 1. 70 and 1*71«
The mathematical e^q^ectation of the minimum endurance value can be deter-
mined by the formula
m,
'^-V^mla ^^^IcffN — AffZ/^vv.
The value of Am^ogN is determined as a function of the mean-square devia-
tion SiogN 3-2^ ^^ "the number of tested ^ecimens, from the curve in Fig. 1. 70.
The mean-square deviation of the minimum endurance value SiogNmin referred
^^ SiogN is given in Fig. 1.71*
Thios, if the characteristics of the distribution of endurance of the /173
structure are known, eq.( 11.15) can be used for determining the distribution of
the minimum endurance values when testing a small number of specimens n. Know-
ing this distribution, 'we can determine the probability of failure of a struc-
ture at a number of cycles of stress of
l&k
(11.16)
where
TIn = reHabiHty coefficient -with respect to the numtier of cycles;
N^i ^ = miniimjin value of the n-umlDer of cycles to failure of the structure
in tests.
where
Taking the logarithm of eq.( 11.16), we obtain
(11.17)
If dynamic tests of full-scale models are carried out at loads equivalent
to the loads acting on the struct lore in question in fUght, it can be taken as
certain that the distribution of endurance in dynamic tests and tinder service
conditions is identical. A difference in these distributions can arise only from
errors in the dynamic tests and from the scale effect in cases in which the
volume of the loaded material in the structure is greater than in the specimen.
An exaiiple would be the case in which a specimen cut out of a blade is loaded in
the test only on its midsection. If dynamic tests are carried out at loads dif-
fering from those acting in flight, then the characteristics of the distribution
of endurance in service substantially differ from those obtained in tests and
can be determined only approximately by a conversion based on specific assunp-
tions with respect to the Wohler curve.
If the distribution of endurance under service conditions
has been
9so r V
deteiTnined, then the conditional probability of failure of one arbitrarily taken
specijnen of a structure in service at the given outcome of dynamic tests ^2 /174
can be determined from the e^^ression
P
con
^^-10$'^/^
= I
.(Si)^?,.
(11.18)
The total probability of failure of this specimen of the structiore in
service will be equal to the sum of the conditional probabilities multiplied by
the absolute possibility of each outcome 9Bin (53)'i52 *
00 L M
dL
(11.19)
After calculating the value of this integral, we can construct the depend-
ence of the probability of failure of the structure P on the adopted magnitude
of the factor of safety with respect to the number of cycles T|n •
If the distribution of endurance in service and in tests is identical
185
^l^N^esT^ ^l^B^s.rv'^ ^^OgNfesT" ^^^gN^erv
then the probaTDility P depends only on two factors: the nimitier of tested speci-
mens n and the ratio of the logarithm of the factor of safety log TIn to the
meanr-square deviation of the logarithms of the niMbers of cycles to failure of
the specimens SiogN (K-g.1.72).
4;n,
'logH_
logN
2 ~y^
1.0
0.5
\
- - - ■
\
^s.
^^
^N
;J
j
w
50 n
10
50 n
Fig. 1.70 Change in Magnitude of
Mathematical Expectation of Mini-
mum Endiorance Values as a Function
of the Number of Tested Specimens.
Fig. 1. 71 Mean-Square Deviations
of Minimimi Endurance Values as a
Function of the Number of Tested
Specimens.
Thus, to determine the required safety factor for the number of cycles, we
can conduct dynamic tests on n specimens of the structure, determine the mean-
square deviation Siog n and N^i^ and, after assigning a certain probability of /175
failure Pasd* derive i\^ from the curves in Fig. 1.72. After this, the safe number
of cycles to failure can be determined from eq.(ll.l2).
^log N
This approach is possible when only few specimens are tested. The value of
can be taken from the results of other tests of similar structures'^.
If we ass-ume beforehand that SiogN = 0.2 (this value is close to the mini-
mum mean-square deviations observed for most helicopter units) and assign the
probabilities of failure indicated in Table I.IS, then values of the safety fac-
^^ A similar approach as related to the calculation of airplane structures was
proposed by V.L.Raykher.
186
p
;
§
^
'^^>:^
^^^
10
1
i
^
N
\
v;n.
/
100
J
^
\
^
^
N
woo
f
^
^
tN
n:
— N
\
nooo
--
^.
\
V
\,
2.^\'>s
\fl\
s
10"^
n
-25--
^\
\
\
\
w
\
^
\
N'
Fig. 1.72 Diagram for Selecting the
Magnitude of the Safety Factor T|n
with Respect to the Nijinber of Cycles
of Stress .
tors close to those given in Table 1.16
can 1)6 o"btained. Usually, the values
of SiogN 3-^e higher. Therefore, the
safety factors obtained "by this method
are larger than those given in the table
(Table 1.16).
The main problem in using the
method presented here lies in defining
the probability of failure of the struc-
ture to be assigned in calculations of
service life. Frequently, the prob-
abilities recommended by different
sources differ by three or four orders
of magnitude [see, for exairple (Ref.43)]-
The values for the probability pro- /176
posed here (in Table I.IS) were selected
with the aim of having them correspond,
with more or less reliability, to num-
bers of cycles smaller than the sensi-
tivity threshold Nq . Therefore, these
probabilities should be regarded as
certain conditional values pertaining
to the normal law of distribution. The
actual values are much lower or even
equal to zero.
9* Determination of S^ogM at Given Fiducial Probability
TABI^ I.IB
As follows from the preceding Subsection, the logarithm of the factor of
safety with respect to the n-umber of cycles log TIn needed for ensuring the given
probability of failure is directly proportional to the mean-square deviation in
the distribution of the logarithms "of
the numbers of cycles to failure of the
structiore S^og ^ . The greater S^o g n ^
the greater should be the factor T|n •
Therefore, the reliability of determine
ing the service life of a structure de-
pending on the admissibility of nimier-
ous adopted assimptions is largely re-
lated with the accuracy of determining
S,
Group of Units
of Different Essentiality
for Safety
Group I
Group II
Group III
Group IV
Probability of Failure
of Structure
1
iqooo
\__
1000
i__
100
\__
10
» lo g N •
Usually, 3-5 specimens of a full-
scale structure are tested to establish
its service life. In many cases it is
considered sufficient to test only one
specimen. There is no doubt that, with
such a small number of tested struc-
tures, there is no possibility for a
sufficiently accurate determination of
187
Illllllllilllll
SjogN* Therefore, it is assimed in the method proposed l)j A.F.Selikhov (see
SuTDsect#8) that S^og jg cannot be determined in all cases. With a small nimiber of
tested specimens, we can take SiogN "based on the test results of analogous speci-
mens of another structure tested earlier. Such an approach greatly simplifies
the process of establishing the service life and proves extremely useful in
practice •
A determination of Siog n "with sufficient reHability is possible by testing
at least ten specimens of the structure. For an estimation of this reliability,
one often loses the concept of fiducial probability distribution of Sjc^gN*
The fiducial probability P is usually selected such that it is possible to
consider confident that the value S^ogN lies in the interval:
where
SiogN = estimation of Si^g n obtained for a limited number of test re-
sults;
q = coefficient greater than unity in magnitude.
It follows from the aforesaid that the unknown value of S^ogN may He with^
in the confidence limits, with a probability p • Consequently, this can be equal
to qSxogN* In this case, the logarithm of the safety factor for the number of
cycles log TIn in conformity with the method presented in Subsection 8 increases
in proportion to the quantity q, which is the reason for the fact that the cal-
culated value of the service life decreases.
The value of the coefficient q depends on the number of tested specimens
and on the adopted value of the fiducial probability P .
Table 1.19 gives the values of the coefficient q and the values of the
fiducial probability P corresponding to them, which we have taken from the book
of E.S.Wenzel »»Theory of Probability** .
As follows from Table 1.19, if - for exatiple - a total of 25 specimens is
tested and the fiducial probability is not less than 70^, then the experimentally
obtained value of Siog n must be increased by a factor of 1.15 when calculating
the service life.
In assigning the fiducial probability, it must be borne in mind that the /177
reliability of determining SjogN should not exceed the reliability of determine
ing all other parameters entering into the calculation of service life. This
pertains primarily to parameters determining the law of distribution of endurance
in the region of small probabilities of failure, such as the threshold q of sen^
sitivity Nq, and to the character of the distribution law itself which only ap-
proximately can be taken as logarithmically normal.
Therefore, the fiducial probability P, characterizing the reliability of
determining Sjog n* can be lowered substantially to values at which the coeffi-
188
I!
cient q will "be not much greater than unity.
Based on these considerations, in determining the factors TIm we often as-
sume q = 1 and use the value of S^og n ^s not being the value corresponding to
the ijpper Umits of the confidence interval at sufficiently high P«
10. Dispersion in the Stress Levels for Various Structural
Specimens and R el iability MarRJn with Respect to the
Aiplitude of Alternating Stresses %
The anplLtudes of alternating stresses set vp in flight in individual speci-
mens of heilicopter parts of identical structure differ considerably.
Actual measurements have shown that,
tudes of alternating stresses differ both
rotor and for blades of different rotors .
sion of the parameters of series-produced
TABLE 1.19
Number
of
Tested
Values of Fiducial Probability /3
in % for Different q
Specimens
1.06
1.1
1.15 1 1.20
1.25
1.3
n=5
14.6
24.1
35.5
46.1
55.6
63.7
n=10
20.8
34
49
62
72.2
79.7
71=25
32.7
51.8
70.6
83.2
90.5
94.4
rt=50
45.2
68.2
86
94
97.4
98.8
in identical flight regimes, the anplL-
for the blades of one and the same
This can be attributed to the disper-
blades because of differences in their
size and shape and hence in their
weight. Usually, there are deviations
from the theoretical contour of the
profile and differences in the geo-
metric twist of the blade. Further-
more, when installing the blades on
the helicopter and adjusting the coning
of the rotor, certain differences arise
in the blade setting angle. All this
ultimately leads to some difference in
the operating conditions of individual
blades and, as a consequence, to a dis-
persion of the amplitudes of alternat-
ing stresses set -up in identical fUght
regimes .
There is also a difference in flight regime parameters associated with the
manner of piloting by individual pilots.
Another difference, which is not smaller but might even be greater, is ob-
served in the stress amplitudes of all other helicopter parts. The dispersion
in stress anpHtude is especially great in conponents where alternating loads
from individual blades should be equal to zero when added (if the blades are
ideally identical), for all harmonics with the exception of harmonics that are
multiples of the number of blades. If the blade parameters are different - and
this is practically always the case - then the small alternating loads with har-
monic frequencies that are multiples of the number of blades in these helicopter
units will be sipplemented by relatively high loads with other hannonics, having
magnitudes proportional to the magnitude of the difference in blade parameters.
The scattering of the values of the alternating stresses in such units may be
very great. Usually, such units include the following: automatic pitch control,
rotor control cocponents, and fuselage parts; in the latter, it is mainly the
reduction gear frame that is especially stressed by alternating loads.
189
To allow for all a'bove factors in calculating the service Hfe, we will /178
introduce the relialDillty coefficient with respect to the anpHtude of alter-
nating stresses Tjcr • This coefficient shouH ensure operational reHaTbility of
any structural specimen in a group of helicopters with consideration of the ex-
isting scattering in the alternating stress values.
Usually, some helicopter is arbitrarily selected for measuring the alter-
nating stresses in a structure. The alternating stresses a^eas obtained in tests
with this helicopter are then used for conducting dynamic tests. This means
that the tests are made with stresses Otest = % cTneas- Therefore, the method
presented a'bove (see Subsect.S) of determining the reliability margin TIn and a
safe Hfe yields results that can be applied only to a specimen of the structure
in which stresses equal to atost ^^® acting. For all other specimens of this
structiore the service life will be longer if the active stresses Qact ^ crteat
and shorter if aact ^ <^teBt •
Let us determine the value of the reliability coefficient TIj from the cor^
dition that the probability of failure of the helicopter xinit in question P^,
with consideration of the existing dispersion in the anpHtudes of alternating
stresses, is equal to the assigned probability Pasd* Usually, this value is
taken to be the same as the probability of failure Fq of the unit in which the
alternating stresses atest adopted in dynamic tests are active.
Thus, if the value of TIn is chosen from the condition that the probability
of failure of the specimen with stresses atest is equal to Pq = ^asd* then the
probability of failure of other specimens of this structure can be determined
by means of the formula
^^2* (11.20)
Here, Pg is the probability of failure of the helicopter unit in which stresses
equal to a are set ip. In this case, the endurance distribution cpg© rv deter-
mined on the basis of djmamic tests for some selected equivalent stress level
which enters eq.( 11.20) should be recalculated with consideration of the fact
that, in different specimens of the structure, different equivalent stresses are
set up.
If we assimie that the endurance changes in accordance with the law
a^N=^consi, (11.21)
then the characteristics of the distribution cps©rv (? ) can be set equal to
/o \ ,o . \LLm22)
{micgN\ = ^lcgN^,A^^^^^t^'^^^^''<^^ ^' (11.23)
190
where
(SiogN)a = mean-square deviation of distribution of the mjmber of cycle
logarithms at stresses Oa^t differing from those used in the
dynamic tests;
(^10 g N )a ~ mathematical ejjq^ectation of this distribution;
c^'tost ~ stresses in the test;
<^act ~ stresses acting in some helicopter specimen.
Let lis assiane that the distribution of the acting alternating stress anpli-
tudes in different specimens of a structure can be taken as logarithmically /179
noraial. Then the probability of failure of the helicopter unit in question will
be equal to
where cpiogCTact ^^ ^^® distribution law of the acting alternating stress ajipH-
tudes (Fig. 1.73).
Here, it must be borne in mind that the value of o^^^^ adopted for dynamic
tests is selected arbitrarily, based on the results of measuring stresses in one
randomly chosen helicopter or in several helicopters. Therefore, the probability
distribution of failure Pj for units with different acting stresses will shift
along the axis log a (see Pig. 1.73) depending on the adopted value of ate at so
that, at Oact = ^test ^ "^h® probability of failure Pg would be equal to Pas^ in
view of the fact that the value of T|n was selected from this condition. Hence,
it is clear that the value of Pnoas ^H depend on the quantity ate at •
Consequently, the probability P^eas ^s a conditional probability for a
specific, randomly selected value of Otest • The total probability of failure of
a unit, arbitrarily selected from a group of helicopters Py,, can be obtained as
the sum of conditional probabilities Pmeas multiplied by the probability of oc-
currence, in this unit, of stresses which had been taken as the basis for the
dynamic tests cpiogg d log a:
00
(11.25)
eo r 00 "1
— 00 |_ — 00 I
If the stresses are measured in one and the same helicopter, then we can
consider that
191
Illlllllllllllllll
Fig. 1.73 Character of the Distribution
of cpioga and Pg for Different Oteat =
If the measurement is made on
several specimens of a structiire 7180
and if, in the dynamic tests, the
following stresses are assigned:
where o^^^g,^^^ is the average anpH-
tude of alternating stresses
measured on several specimens of a
structure, then the distribution
parameters cpioga ^s should be de-
termined as the distribution para-
meters of the average values of al-
ternating stresses
It follows from eq.( 11.25) that the total probability of failtire P^ depends
upon the quantity %. Therefore, after assigning P^ = Paad* ^^ ^^^ determine
the necessary value of Tlj. It is evident that in this case the required value
of % depends on the law of alternating stress distribution for different speci-
mens of identical helicopter units 910 6 0^^.^ ' '^^ determine the characteristics
of this distribution law we can use data from different stress analyses which
are often performed on the same helicopter units in tests made for different pur-
poses #
It is logical that the dispersion of the average equivalent alternating
stresses may differ for different units.
The meanr-square deviation in the distribution of alternating stresses for
different rotor blades usually Hes in the range of
5/,^^=0.02 — 0.035.
If, as is often done, we assume a normal distribution law of the alternating
stress airplitudes, then these values of Sio^g "will correspond to the values
Ya=--^= 0,05 — 0.08,
(11.26)
where
192
Scr = mean-square deviation in the distribution of alternating stress
anplitudes for different blades;
mg = mathematical expectation of this distribution, i.e., average stress
in these blades.
I
For small y^, we can assume Sjoga ~ Ycr ^E ®*
For units whose load depends on the quality of adjustment of the rotor,
such as automatic pitch control, reduction gear frame, and others, the coeffi-
cient Ya is somewhat larger •
It follows from the conposition of eq.(ll#25) that the total probability of
failure P^ depends mainly on two parameters:
b =
r<ycj
where m is the exponent of the Wohler curve.
The total probability of failure P^ depends also on the probability of
failure Pq of the structural specimen with stresses at^st ^ used in the calcula-
tion.
Figure 1.74 shows the calcu-
lations of total probability P^
for different values of a. b, and
Po according to eq.(ll.25} in the
case where the stresses were /181
measured only in one specimen of
the helicopter structure. The cal-
culations were carried out also
for different values of the number
of tested specimens, but it was
found that the total probability
of failure does not depend appre-
ciably on this number. In Pig. 1.74
the broken curves represent ngpec ~
= 5 and the solid curves,
= 20*
n
spo c
= Po =
Fig. 1.74 Results of Calculating the
Probability of Failure with Considera-
tion of Dispersion in the Values of
Stresses Acting in Different Specimens
of a Structure.
If it is required that P^ =
Pasd* then we can obtain
final graphs from which it is easy
to determine the necessary margin
Tla if the values of S^oga, S^ogN*
and Paadi are known. These graphs
are given in Fig. 1.75* As in
Fig. 1.74^ the broken ciorves per-
tain to the case n,
= 5 and
the solid curves, to the case
-spe c
n
Bpe c
= 20.
193
As an exairple, let us find the required margin Tla for a helicopter Dlade if
it is known that Yct = 0.08 (Siogcj = 0.035), and Sios n = 0'4-
First, we determine the value of the coefficient a:
a=~
^logN
0.4
= 0.525.
Then, assigning the value P^] = 1/1000, we olDtain from the curves in
Fig. 1.75:
^^=1.28 ( at n,^^=S)\
from where
^yTia= 1.28*^0.035=0,0448,
and
r]a=l.ll.
If the distrilDution law cpcy is unknown, we usually take Tig = 1.2. This /18.2
value of Tig, as already mentioned in Subsection 7, is often used in practical
calculations .
Rig. 1.75 Diagram for Selecting the ReliaMlity Margin Tig.
11. Method of Determining the Reliahility Margin Tig
Proposed "by A.F.Selikhov
In our presentation, certain methods and arguments differ somewhat from
those suggested by A.F.Selikhov, but the basic principle of the approach to
194
solving the prolDlem is "borrowed from that author •
Here, in determining T|(7 the method descritied in Subsection 8 was used ex-
cept that the dispersion in the anplitudes of stresses acting in flight is taken
into account in the characteristics of the endurance distribution.
If, as "before, it is assumed that - i^^on a change in stress auplitude -
the endurance under service conditions changes in conformity with the law (11.21),
i.e., that
^ogNserr^logNt^^ -{-miiogot^^-^iagc^^^ ), (11.27)
then we can determine the characteristics of the endurance distribution in
service with consideration of the dispersion in the airpHtudes of acting stresses.
The mathematical e^q^ectation of this distribution will be equal to
^X ^ ^hpNiest + ^ (^^9^te,t~ ^%-a^^ ) » ( 11 . 2S )
where m^ogcj is the mathematical expectation of the distribution of stress
anplitudes in different specimens of the investigated structure (the average
value of the ajiplitudes of alternating stresses in different specimens of the
structure) .
If the tests are carried out at stresses of /1B3
au '
then, after putting
we obtain
The mean-square deviation in the logarithms of the n-umbers of cycles to
failure under service conditions can be determined from the formula
If the dynamic tests are carried out at an azrplitude of stresses Otest ^
= l\a Oav (where Oav is the average arrpHtude measured in flight on different
specimens of the structtire), then the dispersion of the characteristics of en^
durance in tests will depend on the dispersion of the acting stresses in differ-
ent helicopter specimens. The amplitude established in tests is a random quan-
tity depending on the results of stress analysis. As before (see Subsect.8),
we are interested in the characteristics of the distribution of the logarithms
of the minimum number of cycles to failure.
195
Let us assume that the values of the ininimuiii numlDer of cycles to failure
obey the law
/ogN^=hgN^,^+m[logo^^^^-^log(ri,G^^)l (11.30)
where
Mg = minimum niam"ber of cycles to failure of a structure, with considera-
tion of the fact that the airpHtude of the tests can he established
as different, depending upon the results of measuring the average
stress atipHtude Oav l
Njij = minimum nijmher of cycles to failure of a structure at a certain
fixed value of the stress anpHtude in the tests o^^g^ .
Then,
If
^2 = ^logN^^^-\'m{!ogo^^,^-log'r\,-mlogo^^y (11.31)
then
We put
°^«/^==^11oWa^^,
lOg<^test=^<^9'^-'\-^^9^o^
iogmo^^^muga
Then, mo = ^iiioeN * ^^ ^^^ value of the meanr-square deviation of the logarithms
of the numbers of cycles in tests will be
^2^/-^/.^^^,„+'^'(V)a.' ( 11.32)
where (Siogcr)av is the mean-square deviation in the values of the average loga-
rithm of the stress anpHtude measured in different specimens.
This value depends on the nxomber of measurements n^^aa •
For one measurement (n^gag = l) /2SL
(«5v)a.=*5/^-. (11.34)
With consideration of eq«( 11.33), the mean^square deviation in the endurance
distribution in tests can be determined 1:^ means of the formula
196
(11-35)
Using the same reasoning as above (see Subsect.S), we arrive at the fact
that the protiatiility of failure in this case can "be determined by an egression
similar to eq«(ll.l9):
P^ =
J ?niln(^2)
£.-/*?^^
J ?serr(«l)^«l
dio
(11.36)
If the distribution of the logarithms of the minimum number of cycles to
failure in tests can be represented approximately by the normal distribution law,
then eq.( 11.36) can be rewritten in the form
23t 1/ ^2
1 1
2 si
U-log-nN
J 5i '
25^
dli
dU
(11.37)
If we introduce new variables
;,=JlzL^a„rf|- J2:z^
St. S2 '
(11.38)
then eq.( 11.37) is transformed into
/ (12) ^I
2 ^r
e ' dl,
d\.
(11.39)
where the ipper limit f(5 2) is determined by the e^q^ression
(11.40)
Substituting here the values of mx and m2, we obtain
(ii.a)
It follows from this eaqjression that the probability of failure P^ can be
determined for each Tla, if we know the values of
197
Sly S2, ryvond^m;^
'N
''^^logNtesf-^IogN^,^'
We can propose the following method of determining the required margin TlgS
First, construct the dependence of Sg/Si on ^ "^^^ ^^ '*' "^^^ '^^ '^ ^iq^n for
Si
assigned values of Fj^ (see Fig.l.76)# Then, after determining Si and Sg from /2S5
eqs. (11.29) and (11.35), use Fig. 1. 76 for determining the assigned value Fg,^^
Si
(11.42)
from where, knowing TIn, Amiog n (s®® Fig. 1. 70), and Si, it is easy to determine
also Tier.
The method proposed here is rather sinple, although it involves somewhat
more conplicated conputations for determining the service Hfe in coirparison with
the method proposed in Subsection 10,
0.6
OA
az
/
/
7
\
1
1
/
/
/
/
i
/
i
/
/
/
1
/
1
'/
]
1
1
' -
1
1
I
'-i
^^m
^. mo
'-.«
1
000
Fig. 1.76 Graph for Determining
m log % + log TIn + AmiogN
as a
Function of S1/S2, for Different
Assigned Prohabilities of Failure.
steel spar.
where the margins T|g were calcu-
lated on the "basis of values taken
directly from the graph.
It follows from the above for-
mulas that, under the assumptions
adopted here, the margins of reli-
ability TIn and \^ can be combined
into a single criterion T| = TInTIj or
one reliability margin can be sub-
stituted for the other. This is
convenient in carrying out calcula-
tions and conducting dynamic tests,
a fact already mentioned in Sub-
section 7, but it offers no advan-
tage in selecting the margins T|n and
T|cT since their values are determined
from different conditions.
12* Example of Calculation of
Service life
As an exanple, let us calculate
the service life for a blade of a
he.avy helicopter with a tubular
In determining the service life of a blade, the calculation is first per-
formed for sections located at different relative radii, after which the service
life obtained for the weakest section is established for the entire blade.
198
let the weakest section be that at a relative radius r = 0.74*
We now assume that the results of the dynanac tests of five specimens of
the spar at an alternating stress anplitude ±15 kg/mm^ are as follows (see
Table 1.20):
From tests, we draw the conclusion that the endurance Umit a^ of the /186
specimens Nos«2, 3, 4, and 5 is higher than a^ = 15 kg/mm^. Consequently, the
probability P that the endur-
ance limit Ow is lower than
TABLE 1.20
15 kg/inn can be taken as
equal to 0.2«
No.
of Specimen
Number of Cycles
of Stress
Test Results
No.l
9,8x106
Sipecimen failed
No. 2
2Q^\0s
No. 3
No. 4
20^*106
20>'106
Specimens did not
fail
No.5
20*106
On setting Sj
equal
to 0.07 (see Subsect.l3), it
will be found that the endur-
ance limit a„ =13 kg/mm^
corresponds to a 5% probabil-
ity. This limit will be con-
sidered minijntim.
The margin for the nimiber
of cycles can be taken either
on the basis of practical e^^^erience by assigning the service life in accordance
with Table 1.16 or on the basis of the method of A.F,Selikhov (see Subsect.8).
Based on Table 1.16 for Group II (blade equipped with a spar-damage warning de-
vice) and n = 5, the reliability margin TIn can be taken as equal to about 2.7*
In the second case, SiogN must be known. It is obvious that merely from
the results of tests it is inpossible to determine the value of Sjo^m . However,
it is possible to assign a certain value to S^ogN based on results of tests with
similar specimens.
Let us put SiogN = O./f. Then, assigning the value F^.^^ = 1/1000 (GroigD II
of units) we obtain log TIn = 2-3 * SiogN* i.e., T|n = 8.3, from Fig. J. 72. Thus,
the required reliability margin for the number of cycles T|n according to Seli-
khov's method is substantially greater than that obtainable from service life de-
terminations. In many cases, this difference is partially coirpensated by intro-
ducing the concept of fatigue limit into the calculation and by refining the re-
quired reliability margins %.
As mentioned above, in defining the service life with the use of the reli-
ability coefficients selected on the basis of practical e^q^erience, the value of
Tier can always be taken as equal to 1.2- However, this coefficient can be re-
fined in conformity with the method proposed in Subsections 10 and 11. For this
purpose, more conplete data are necessary on the dispersion of the alternating
stress anplitude for different specimens of the structure.
Let us assume that the stresses are measured in only one specimen. However,
on the basis of experience in meas;iring similar units of other helicopters it can
be assumed that Ycr = O.OS and thus Sioga = 0.035* Then, by means of the methods
199
presented in Subsections 10 and 11 we obtain Tlcj = l.ll. Nevertheless, we will
take % = 1»2.
The minimijin value of N,,!^ of five tested specimens (n = 5) at an alternating
stress airplitude of a = Tl5 kg/irni^ is \ij^ = 9*8 ^ 10^ cycles.
The number of cycles corresponding to the minimum fatigue Umit is deter- /187
mined from the formula
while the values of Mj are obtained from the formula
In the calculation of service life, we will assume that, in all regimes
where the acting stresses are below the minimum value of the fatigue Umit^ no
increase in damage potential for the structure takes place.
We will not calculate the equivalent stresses in individual flight regimes,
but will assimie them as equal to the maximum measured stress anplitudes. In
this case, the value of e^ will be either zero 'or unity.
The calculation of equivalent stresses is given in Table 1.21.
If we assume that the endurance obeys the law (11.21) at all alternating
stress levels and that there is no fatigue limit (in this case Si = 1 in all re-
gimes), then all flight regimes are equivalent in damageability to the regime
with a stress anplitude of aeq = 11.5 kg/mm^ acting dioring the entire service /188
life of the blade. In this case.
If it is assumed that the minimimi endurance liinit is a- = 13 ke/mm^,
then one regime [see eq.(ll.9)3 with an anplitude o^^^ = 13.6 kg/mm^ will be
equivalent to all flight regimes. The duration of this regime, as follows from
Table 1.21, will be about 23^ of the service life of the blade (e = 0.229).
Then the lifetime itself can be determined in the following manner:
200
N^^a = N,
A^n
R--
60/e
-A29 /irs.
The same results can "be obtained from the total damageability without mak-
ing use of the concept of equivalent stresses [see eq.(11.5)]:
Ns =^
.3.09x106;
/?=■
60/
-A29hrs.
TABLE 1.21
EXAMPLE OF CALCULATION OF BLADE SERVICE LIFE WITH RESPECT
TO A SECTION OF RELATIVE RADIUS r = 0.74
Flight Regime
0.1
6.0
8.8
£=0,5;
^.•<^L.
l^i
a/
Hovering
9.7
11.64
oo
Low speeds
K==20 A://7/hr
1
0.03 7.2
10.5
11.6
13.92
1.84*105
0.016x10-6
K=30 km/hr
1
0.02 10.5
13.2
15.02
18,02
0.39-106
0.051*10-6
V=^60 km/hr
1
0.05
12.4
12.5
15.05
18.06
0.39x106
0.128x10-6
Takeoff
1
0.02
9.5
12.4
14.0
16.8
0.59x106
0.034?ilO-6
CJimb
0.06
6.0
5.6
6.9
8.28
oo
Cruising speed
0.55
8.0
9.0
10.5
12.60
oo
Maximum speed
1
0.10
8.0
10.5
11.09
13.31
2.4x106
0.042x10-6
Gliding
0.05
7.5
7,2
8.8
10.66
Ort
Braking
ist stage— <J„,ax
1
0.002
15,2
18.4
21.11
25,33
0.05)fl06
0.04)410-6
2nd stage— 0.7 o„,ax
1
0.007
10.64
12.88
14.79
17,75
0.43x106
0.016x10-6
3rd stage— hovering
c=0
0.011
.229
6.0
8.8
9.72
11.66
oo
2 0.327x10-6
201
These results show that on introduction of the concept of fatigue limit or
endiirance limit, the service Hfe of a blade will be greater when derived from
calculation.
However, it must be borne in mind that the margins with respect to the num-
ber of cycles presented in Table 1.16 were introduced into the calculations with-
out assumption of the existence of a fatigue limit. Therefore, they should not
be used in calculations with a fatigue limit.
13 • Possible Ways of Determining the Minimum Endurance
limit of a Structure
The above exanple indicates that substantially higher values for the service
life of a structure can be obtained when making use of the concept of minimum
endiurance limit. Therefore, a determination of these values is mandatory in
many cases.
A sufficiently accurate determination of the values of a,
m i n
from the re-
sults of tests is virtually infeasible. Only a highly approximate determination
of this value is possible. Even then, an appreciable increase in the number of
test specimens is required* Nevertheless, in calculations of service life, even
approximate endurance limits will closely approach the calculation results to /189
reality and offer the possibility of developing more competent technical solu-
tions. Therefore, it- is always advisable to resort to a determination of endur-
ance limits, using both approximate and siirply foiinal methods of calculation.
P%
99-
9S
90
80
50
10
10
5
1
OJ
0,01
<^/ai
' -^
~u
r-
1
.
1
m
L- .
\y
^
r-T
-
iOfS
s
^
Inio \
^
—-
r-^.
k"*r~
^^min
-—
y
<_^
^W
*-
\ ^
f
_..
\yt
\--
-^
i?^
/
:7^
r'
— -
-
^H
J«,'
= a
065
-
.''
1
I _
-
1
^
\
i
%
0,9
0,5 0.6 0,7 0.8
Fig.1.77 Distribution of Endurance limits.
Primarily, an attenpt must be made to define and determine the parameters
of the distribution law of endurance limits. Toward this end, fatigue tests
must be performed with specimens at several alternating stress levels, located
in the region of endurance limit distribution. The tests should be carried out
202
IB II HII I I ■IIHHH
on the "base of a sufficiently large number of cycles • In selecting the test
base, it is generally assianed that, for steel specimens, this base can be set
somewhat greater than 10''' cycles, for example 2 x 10''' cycles, whereas for d;aralu-
min specimens the base must be somewhat higher than 2 ^ lO''' cycles (frequently,
a base of 5 ^ lO''' cycles is used).
The probability that the endurance Umit is higher than the assigned al-
ternating stress level is defined as the ratio of the number of specimens tested
at the given base at no failure n^o^an ^^ "the total number of specimens tested
at this and at a lower level of stresses n:
The resultant distribution of endiirance limits can coincide with the normal
law only in a small section corresponding to the average values of probability
(i^g.l»77)- At small probabilities, the distribution of the endurance limits
deviates from the noimal law and has a certain sensitivity threshold a„ • At
tn i n
large probabilities, beginning with some stress Of an , all specmens fail with-
out having been subject to the assigned base of the test.
The distribution of endurance limits at average probability of failure is
best represented by the lognormal distribution law. This can be used also for
determining the minimum endurance l±mit.
Available results of tests on blade specimens show that, for this law, we /190
can take values of S^oga equal to about
S/o^.^ =0.05 — 0.07,
where SiogCT„ is the mean-square deviation of the distribution of fatigue limit
logarithms .
It is impossible to propose a sufficiently reliable method for determining
a„ . Thus, we can suggest only a purely formal method which, however, yields
m 1 n
sufficiently good results in practice. It can be assumed that the minimum en-
durance limit coincides with the value of a„ corresponding to ^% probability of
a logarithmically normal distribution law of endurance limits.
If such an approach is used, the values of o^ can be refined by a method
in which fatigue tests are carried out at two alternating stress levels, close
in anpHtude. The test specimens, at least 15 - 20 of them, are divided into
two groups.
The first group is tested at maximTmi alternating stress which sipposedly
does not exceed the minimum endurance limit; for this reason, it is desirable to
prevent any of the specimens from failing at a number of cycles corresponding to
the selected test base. The results of testing this group serve to confirm that
the minim-um endurance limit may actioally correspond to their test level.
203
The second groip of specimens is tested at somewhat higher alternating
stresses, so that a certain percentage -will fail without having operated the
rated niMber of cycles. After determining the probatiility that the endurance
limit is iDelow the anplitude of the second test level and after assigning some
value of SiogCT„, we calculate the value of a„ corresponding to the ^% probaTDili-
ty. If the test data of the first grouj) do not contradict this result, then the
resultant value of a^s% can "be taken as the minimum endurance limit.
Occasionally, it is assumed for greater reliability that the minimum endur-
ance limit corresponds to smaller probaiDility values, say, a protiability of
1/100 • However, it seems that still lower values of this probability are not
advisable.
It should be noted that in Tnany cases an arbitrary concept, which could be
called the reduced endurance limit, is used for characterizing the fatigue
strength.
The reduced endiorance limit is determined by converting the test results,
by means of eq.(11.3)^ to an arbitrary base which often is taken as N^^^g^ =
= 10*^ cycles for steel and N^g^g^ = 2 >^ 10*'' cycles for duraliomin:
where
cjteflt ^ alternating stress arrplitude in the test;
Np = number of cycles to failure corresponding to a probability of
failure equal to P;
m = exponent of the Wohler curve, usually taken as m = 6.
If we take Np corresponding to the probability of failure as equal to 5%, /I9l
then the value of o^t^^^ furnishes an approximate idea as to the magnitude of the
minimum endurance limit. The minimum value of the number of cycles to failure
of a given structure \^^ often is substituted for Np as the characteristic of
fatigue strength. We must enphasize that the reduced endurance limit, irrespec-
tive of the manner in which it is determined, does not correspond to the concept
of endurance limit in the sense in which it is used above, in this Section.
It is also of importance that the distribution of the reduced endurance
limits has a mean-square deviation equal to
which is almost always greater than the value of Sioga^*
204
14* Advantages and Disadvantages of Various Approaches in
DeterminiriA^. the Necessary Reliability Margins « and
Estimat ion of their Accuracy
The sinplest approach, as already shown a*bove (see Subsect*?)* is to calcu-
late the service life under application of the coefficients Tlf^ and Tlcy taken on
the basis of practical e^g^erience in defining the service life. These coeffi-
cients have been checked on a large nimiber of helicopters and many hundreds of
units have successfully lived out the service life thus established • However,
it must be borne in mind that the use of the coefficients T|im and Tla has been con-
finaed by practice only in combination with some method of calculating the
sei'vice life which, in particular, differs by the following assumptions:
1. No endurance limit exists, and the Wohler ciirve is described by
eq.(ll«2l)- Accordingly, the coefficients e and e^ are taken
as equal to unity.
2. In each flight regime, the stress airplitude is considered equal to
its maxim'um measured value in this regime.
However, such an approach to the calculation of service life has substan-
tial shortcomings:
1. In determining the service life, one disregards the difference in the
dispersion of the characteristics of endurance which may be dissimilar for units
of different design which, furthermore, are dissimilar with respect to the ma-
terials used and the manufactioring process. The magnitude of dispersion of
stresses acting in different specimens of a given structure is also disregarded.
2. Rejection of the concepts of endurance limit and exclusive use in the
calculation of the maximum stress amplitudes in each flight regime lead to in-
correct ideas as to the share of damage potential contributed by different flight
regimes .
Therefore, an attenpt should be made to use in^roved methods, incorporating
the basic principles of the theory of probability. One of the possible variants
of this approach is given in Subsections 8, 10, and 11.
It is necessary to point out that this method, in the form in which it is
presented here, gives conpletely satisfactory values of service life rather
close to those obtained by the preceding method. Of course, there is some re-
distribution in the values of the safety factors. The margin TIn is substantial-
ly larger whereas the margin T|ct is smaller. Furthermore, the concept of mini- /192
mum endurance limit should be used in the calculation. Otherwise, the service
lives will be underestimated.
In appl^ng this method, such large probabilities of failure (equal to
1/IjDOO or even more) often raise doubt. Actually, this means that one unit out
of 1000 should fail din'ing its rated service life. Therefore, we must again em-
phasize that the indicated probabilities are purely conditional values, corre-
sponding to the normal distribution law of endurance. In reality, in the •region
of small values of probability of failure, this law deviates from the normal and
a sensitivity threshold is observed in the endiorance characteristics. Its values
lie in the probability region of about l/lDO or fluctuate about this value. Conr-
205
sequently, assignment of a conditional prolDability of 1/1000 is actually equi-
valent to the requirement of a very small or even zero probability. Therefore,
we cannot agree with those authors who are not afraid to stipulate a probability
of the order of 10"^ or even ICT'^ , under application of the noimal law of dis-
tribution of endurance. There is no sufficiently valid reason for such demands.
Generally, anyone familiar with the above method will object to doing away
with refinements of experimentally obtained values of the mean-square devia-
tions SiogN, based on the rather high values of fiducial probability accepted
in practical applications of the probability theory. If such a refinement is
made, the calculation would have to incorporate a two-fold value of Siog^ (see
Subsect.9)* which would lead to an increase in the required margin T|n and thus
to a decrease in service life.
In addition to the above considerations (see Subsect.9)j another inaccuracy
in the proposed method of calculation should be pointed out. Usually, the
equivalent stresses acting in different flight regimes are replaced by their
maxim-urn values, leading to an londerestimation of service life. These two inac-
curacies mutually cancel out, and an elimination of one should definitely be ac-
coirpanied by elimination of the other. In such a case, the values of the service
lives obtained by calculation do not change substantially.
There is no doubt that in time, as new e:5q:>erimental data are collected,
more extensive refinements will have to be introduced into the method of calcu-
lating the service life. Practical experience in operating helicopters and the
ever greater number of results of dynamic tests will also furnish an incentive
in this direction.
15. Blade Strength Recruirements in Design Selection
A helicopter blade operates under conditions severely taxing its strength.
During its entire service life, the blade is subject to excessive static and
variable loads. This characteristic of the blade operating conditions iuposes
extremely stringent requirements on its structure and primarily on the fatigue
strength of its main element, the spar. Consequently, the blade spar should be
made only of materials with high fatigue strength characteristics.
Blade designs with tubular steel spars and pressed duralumin spars are /193
the most common type at present.
Excellent results can be e3q)ected when manufacturing spars of various syr>-
thetic materials. Blade designs with a glass-laminate spar are already known.
However, practical experience operating such blades is still insufficient. For
this reason, we will not further discuss the strength aspects of such types.
The most important requirement for blades with steel and duralumin spars
is that of maxim-um elimination of any stress raisers which lower the fatigue
strength. The use of bolts and rivets is inpermissible in blades. The frame of
the blade is fastened to the spar exclusively by glued joints.
Fittings with large stress raisers can be tolerated only in segments with
206
INI
small alternating stresses, for exairple, in the blade root close to the hulD
hinges • In this case, despite the small alternating stresses, the section of
the spar near the root joint must "be increased by a factor of 3 - 4» Only a
very appreciable reduction in alternating stresses will permit the use of fit-
tings with stress raisers •
Fatigue strength is also distinctly lowered by small technological defects
which also act as stress raisers • Consequently, in the manufacture of blade
spars the process used must be aimed at conplete elimination of all apparent de-
fects of the spar.
To eliminate the possibility of some flaws remaining undetected, the spars
must be subjected to rigorous inspection under application of all modern methods
of nondestructive materials testing.
Below, the strength properties of a blade with steel and duralumin spars
will be investigated in greater detail.
16. Strength of_a Blade with Tubular Steel Spar
Cold-rolled tubing of high-alloy steels 30KhGSA or 40Kh]MA quenched and
teirpered to a strength of o^ = 110 - 130 kg/mm^ is commonly used for the blade
spar.
After hot- and cold-rolling, shaping, and quenching, the outer and inner
surfaces of the tube are poUshed. Recently, cold-working of the spars has be-
come a mandatory operation after polishing.
A thus manufactured spar without cold-working may have a minimum endtorance
limit of the order of a^f^ ^ =12-13 kg/mm^ at an average coirponent of the
cycle CTu, = 20 - 25 kg/mm^ . However, the strength is reduced greatly if, in
manufacturing the spar, various technological defects and miscalculations are
permitted. The following can be mentioned as the most dangerous types:
In ternal cracks and laps . During hot-rolling, plastic deformation may be
accoirpanied l^ partial tearing of the material. This usually occurs at a re-
duction in tenperature of the workpiece during rolling and also as result of con-
tamination of the steel by nonmetallic and gas inclusions, the formation of
films, high porosity, segregation, and other metallurgical defects. The cracks
run into the workpiece at an acute angle, so that it is often difficult to trace
the outcropping of the crack on the surface.
On further cold-rolling, the degree of defoi«mation increases and the crack
folds over into the wall of the tube at an ever smaller angle to its surface. /19.4
Usually a series of such internal cracks is observed. They are small, being'
about 0.1 - 1.0 mm deep and 3 - 10 mm wide.
Laps appear vpon cold-rolling on the outer surface. They are usually due
to extensive surface roughness after hot-roHLng. Subsequent plastic cold-work-
ing leads to an uneven flow of the material during which defects known as laps
207
and seams may form. Laps are also atile to form lay flow of metal into the gap
iDetween the roll grooves and formation of a fin which folds over i^Don subsequent
deformation #
Both defects can "be detected by magnaflux inspection of the poUshed sur-
face # Figure I.7S shows characteristic internal cracks at the inner surface of
a spar# The micrograph was obtained dur-
ing magnetic inspection. The endurance
Umit of a tube with seams and laps
drops to a^ , =5-7 kg/mm^.
Fig. 1.78
Cracks on Inner Surface
of Steel Spar.
Rolled-in scale on inner surfa ce.
After hot-rolling, a layer of scale is
left on the tube surface, which has a
greater hardness than the metal. An^
nealing is done after each step in cold-
rolUng. Although annealing proceeds in
an inert atmosphere, thin films of scale
are formed on the surface, due to the
oxygen content of the metal. If the
scale is not conpletely removed, it will
be crushed during the rolling process
and forced into the metal, forming so-
called rolled-in scale. On the exposed
outer surface of the tube, the rolled-in scale is readily eliminated by machin-
ing. On the inner surface of the tube whose machining is more complex and pos-
sible only by belt-grinding or hydraulic poUshing, the rolled-in scale cannot
be conpletely removed. Therefore, small but acute-angled pits of a size not ex-
ceeding 0.1 - 0.05 mm and difficult to detect during inspection, may be left
even after grinding. The fatigue strength of the surface drops in this case to
Ow _ =10-12 kg/mm^.
Din
Rolled-in scale can be eliminated by turning and grinding the surface of
the workpiece after hot-rolling until all scale is removed and by sandblasting
after annealing before each operation of cold-rolling.
For a complete elimination of cooling cracks, laps, rolled-in scale, and
other surface defects, longitudinal grinding of the outer and inner siorfaces of
the tube, after final cold-rolUng and before shaping, is highly effective.
Reduction of fatigue strength from tube, straight enin^«> After quenching and
tenpering, the spar tubes are bent sUghtly. Therefore, before assembling the
blade, the tubes may need straightening. This sets -up residual stresses in the
tube material. Usually, Umiters are used during the straightening operation,
to keep the residual tensile stresses in the tube from exceeding 10-20 kg/mm^.
These stresses increase the average conponent of the cycle and lead to a decrease
of 20 - 25% in the endurance Umit. Still greater reductions in strength may /195
occur if the straightening is inproperly done. To do away with the necessity of
straightening, the quenched tubes should be tenpered in special devices that
eliminate the strains produced on quenching.
In estijuating the fatigue strength of spars, special attention must be paid
2D8
to the possibility of fretting corrosion. Fretting corrosion is an almost cer-
tain attendant phenomenon of cyclic "blade stresses and leads to a substantial
reduction in fatique strength. This usually occurs at points where there is
mating between parts and the spar, if relative microslip is present between the
osculating surfaces. Points of clanp installation
for attachment of the blade frame are the usual
seats of fretting corrosion in steel spars.
Figure 1.79 gives a micrograph of a rijptured
spar. The root of the fatigue crack coincides with
the seat of fretting corrosion.
A marked increase in the dynamic strength of
steel spars can be obtained by mechanical work-
hardening of their surface, known also as cold-
working .
At present, cold-working of spars has become
an almost indispensable operation in the fabrica-
tion of blades. Three methods of mechanical
strengthening have become common in helicopter en-
gineering: the dynamic method of M.I.Kuz'min, the
vibratory impact method of S.V.Ochagov, and the
shot-peening method. The choice of the method gen-
erally depends on the characteristics of the struc-
tural conponent to be strengthened and on the pro-
duction facilities. When using the dynamic method
for strengthening the outer siorface of a spar, its
inner surface is work- hardened by shot-peening. In
devices for the vibratory iirpact method, main enphasis is
treatment of both inner and outer siorfaces of the spar
Fig. 1.79 Incipient
Fatigue Failure from
Fretting Corrosion.
developing coirplicated
usually on simultaneous
by this method.
An increase in fatigue strength is obtained by older methods of cold-work-
ing. The best method, giving the most stable results in treating the outer sur-
face of steel spars, is M.I.Kuz'min's dynamic method.
The increase in fatigue strength due to cold-working is attributed mainly
to two causes: The outer surface of a given part which is most sensitive to in-
cipient fatigue failure is rendered smoother (Fig.l.SO) and residual conpressive
stresses are set im in the surface layers which, in conformity with Hay's dia-
gram (see Fig. 1.63), leads to an increase in fatigue strength of the surface
layer of the part.
Figure 1.81 shows the distribution of internal stresses in the material of
a steel spar, obtained by dynamic cold-working and grit-blasting. Grit-blasting
sets vp almost the same residual stresses as the shot-peening method of cold-
working.
The increment in fatigue strength due to coM-working is especially large /I96
in the presence of fretting corrosion. Apparently, conpressive stresses inpede
the spread of corrosion into the material. Figure 1.82 shows the results of
209
testing steel spars with cold-worked and noncold-worked surfaces operating under
conditions of onset of fretting corrosion.
M If 000-1
n
MkQO'1
VTTTTT??,;,,;,,^;^;^/;^^
6 kg/mm
20
10
'10
'-ZO
'30
'W
-50
'60
-70
h)
M^QO: 1
/^
:^
("1""
CfOTS>-
— ~
/
\0J 0,1 0.3 DM' 0.5 0,6 0.7 '
3.3 J.* 3.5 bmm
I
/
1
!
—
--
'
—
—
—
—
1
1
—
—
1
N
^3^
f
1
I
Fig. 1.80 Surface Profilogram of
Spar Pressed from Aluminum Alloys
after Machining (a) and after
CoH-Working (b) .
Fig. 1.81 DistrilDution of Internal
Stresses from Cold-Working with
Respect to Wall Thickness of Tubular
Steel Spar.
Cold-working by the method of
M.I.Kuz'min;
Triple grit-blasting.
9%
B3\
30-
80'-
10'
60-
50-
^0-
30l
20-
10
1
1 T :::.. , :" "^
"t: ::"""""""
a b
r ^ ,
1
r ^ 1
_4__ l_ f_ __ \.
o
\ ; t^ .„ . _L .LI tt.hKt J 111 1 1 1 . 1 .
•A^ p-|-} ^
— 1 — ii*p - ,.-j_ _^ —1- -, , — -
^=^-t---^-i
-i-J — ^-^ j -1- p!--];---'
-
^i ^ ~ -i- 5 i-
•?
J 1 -. - _. L
10 11 1^ 16
18
10 31 3U 36 38 hO a^ kglmm^
Fig. 1.82 Distribution of Reduced Endurance Idmits of
Tubular Steel S|pars under the Effect of Fretting
Corrosion,
a - Surface polished and sandblasted; b - Surface polished and
sandblasted three times with grit; c - Surface cold-worked by
M.I.Kuz'min's method.
210
The fatigue strength of steel spars can be increased tsy cold-working by a
factor of 1#5 - 2 and, in the presence of fretting corrosion, "by a factor of
2*5 - 3.
Cold-working will raise the fatigue limit of a steel spar to values of the
order Oy^^^^ = 28 - 30 kg/mm^ at a„ =20-25 kg/ran^. Thus, cold-working has
proved a nfest effective means of increasing the reliability and service life of
"blades •
17 • Strength of a_ Blade with Duraltjmin Spar
7197
The most in^Dortant problem in designing blades of this type is to secure a
siofficiently high fatigue strength of the spar. Generally, attachment of the
frame to the spar is accomplished by
glue and thus creates no substantial
stress raisers in the spar. Stress
concentrations in spars are due mainly
to small defects tolerated in its
fabrication.
The s\rrface finish of a spar
plays the main role in reducing its
fatigue strength. A milled and sand-
blasted spar made of AVT-1 alloy with-
out machining of the inner surface may
have an endurance limit of the order
of a„^^ = 3«8 " k*2 kg/mm^ at an
/**H^' ^_!-' .
:>..
mi
average conponent of the cycle a^
= 6 kg/mm^.
Fig. 1.83 Microsection of Spar Wall
through Blowhole Formed in Pressing.
The fatigue strength of a given
spar may be reduced due to defects
produced in its pressing and machin-
ing.
Frequently, the inside channel of the spar is not machined after pressing.
Therefore, pressing defects may remain on the inner surface: adherent metal
slugs, longitudinal scratches, blowholes (Fig. 1.83), and, finally, coarse-
crystalline rings. These defects may reduce the fatigue strength to values of
= 2.5 - 3.0 kg/mm^ (a^ ^ 6 kg/mm^). This suggests to follow the pressing
"^min
by machining of the surface of blade spars with relatively high stresses.
A substantial reduction in fatigue strength is produced also by nonmetallic
and gas inclusions. To eliminate such inclusions, a special melting practice
should be used (settUng of the metal, teeming from certain levels, filtering
through mesh filters, etc.). The best metal is obtained by melting in electric
induction fiornaces, with holding of the molten metal in electrically heated
mixers.
To elbninate the possibility of overlooking nonmetallLc and gas inclusions.
211
each spar should "be suTDJected to careful ultrasonic inspection.
No less inportant is the elimination of possible corrosion pitting of
pressed spars during fabrication (as well as under service conditions )• Practi-
cal" experience has shown that surface and intercrystalline corrosion of a depth
to 0.1 - 0.15 nun will greatly lower the endurance limit. Therefore, metals of
high corrosion resistance should be used for "blade spars, and special measures
must be taken in fabrication to protect the spars from corrosion by electro-
plating after intermediate treatment steps (for exanple, anodizing).
6 kglrnvj^
Q OJ 0.1 03 OA
/198
0,6^1
h)
RLg.1.84 Distribution of Reduced Endxirance limits
(to a Base of 10 '^ Cycles) of Pressed Spars Made
of AVT-1 Alloy with PoHshed (Circles) and Cold-
Worked (Crosses) Surfaces (a) and Distribution
of Compressive Stresses in the Thickness of the S|par
Wall from Cold-Working by S.V.Ochagov^s Vibratory
Impact Method (b).
A marked increase in fatigue strength of spars made of alumin-um alloys can
be achieved by cold-working of the spars. Figure 1.84 gives the results of
fatigue tests of cold-worked spars conpared with spars without cold-working. The
distribution of internal stresses set xsp by cold-working is also shown. The en-
durance limit of cold-worked spars can be raised to values of aw_ _ = 5*5 to
6.0 kg/mm^ (a„ =6.0 kg/mm^) .
m 1 n
It should be noted that the strength of cold-worked duralumin spars is re-
duced greatly if the spar frame, during the gluing process, is heated to a tem^
perature of about 200° C and higher. This makes it mandatory to control the tem^
perat^are in the gluing operation.
18. Effect of Service Conditions on Fatigue Strength Qf_.S pars
The above method for determining the fatigue strength and service life can
be used only if the structure, during actual service, does not suffer mechanical
or corrosion damage. Otherwise, the approach to determining the service life
must be modified and reduced to a study of the effect of such damage. From this
212
vieitfpoint, the stinictures of all blades should be divided into two types: "blades
with protected and blades with e^qDOsed spars .
In a blade design with tubular steel, the spar is usually conpletely pro-
tected by the frame and cannot be mechanically damaged in service. The greatest
risk in such a design is corrosion; therefore, the service Hfe of such blades
is determined by the quality of the anticorrosion coatings of the spar.
In blade designs in which the spar forms the contour of the leading edge of
the profile, special attention must be paid to its protection from mechanical
damage. If such protection is inadequate, the service life is shortened and /199
becomes dependent on the degree of damage of the spar. Usually, a permissible
degree of damage is stipulated here and checked during pref light blade inspec-
tion.
To estimate the effect of damage of a spar in service, dynamic tests are
run on specimens cut out of blades operated for a certain number of hoiors under
various service conditions, followed by establishing a rated service life based
on the conditions of endurance of specimens undamaged in service. lA/hen the
fatigue strength decreases excessively, measures are taken to improve the pro-
tection of the spar.
213
CHAPTER II
HELICOPTER VIBRATIONS
/20Q
Section 1. Forces Causing Helicopter Vibrations
1. Excitation Frequencies
Since, in forward flight of a helicopter, the rotor blades which are suId-
ject to the effect of time-variant aerodynamic forces vibrate both in the plane
of rotor thrust and in the plane of rotation, the reaction forces acting on the
blade in the hub hinges are also variable in
time. Correspondingly, variable forces equal
in magnitude to these reaction forces act on
the rotor hub.
The variable forces acting on the rotor
hub and produced by the vibrating blades can
be given in the form of three forces X(t),
Y(t), Z(t) and three moments relative to the
coordinate axes M^Ct), My(t), M^Ct) (Fig. 2.1).
If the helicopter has an antitorque rotor, the
blades of this rotor will cause time-variant
forces of the same origin to act on the heli-
copter; these can also conveniently be given
in the form of three variable forces and three
moments .
Fig. 2.1 Forces and Moments
from the Rotor, Acting on a
Helicopter.
The variable forces from the vibrating
rotor blades, acting on the helicopter, are
the main soiirce of fuselage vibration.
Fuselage vibrations may also be caused directly by aerodynamic forces act-
ing on the fuselage due to the fluctuating airflow repulsed by the rotors. Thus,
the velocity of the flow pushed back by the rotor in the fuselage region in^
creases whenever any of the rotor blades passes above the fuselage. However,
n-umerous calculations and measurements of pressure fluctuations at the fuselage
demonstrate that these variable aerodynamic forces are appreciably weaker than
the variable forces produced by the vibrating blades and acting on the rotor hub.
For exanple, for the Mi-4 helicopter the variable force acting on the fuselage
due to fluctuations of the flow repulsed by the rotor in the most unfavorable
flight regime (deceleration before landing) is of the order of d=10 - 15 kgf ,
whereas the variable forces acting on the rotor hub in different flight regimes
are of the order of ±(200 - 600) kgf. Therefore, in analyzing helicopter vi- /201
brations we are primarily interested in variable forces inposed on the rotor hub.
These forces, generally speaking, can be defined as dynamic reactions at
forced blade vibrations in flight, for which the calculation methods are pre-
sented in Chapter I. Here, it must be enphasized that the variable forces in
214
such a calculation are determined "with consideralDle inaccuracy. The reason for
this lies in the fact that, in calculating "blade vi^brations, only the lower har-
monics of the loads are satisfactorily determined and the calculation errors in-
crease with an increase in the order of the harmonics. Furthermore, as will he
shown "below, in the calculation of vibrations it is the high harmonics of exci-
tation that are of decisive inportance. This is due to the fact that all methods
of vibration analysis presented in this Chapter are of a mainly qualitative
nature.
An exact calculation of vibrations by means of methods presented in this
Chapter is possible only in certain special cases. The most inportant of these
is the designing of a new helicopter fuselage or
even of a helicopter of a different configuration
1- (for exairple. tandem or side-by-side in place of
single-rotor) equipped with previously used rotors,
7j for which the variable forces were determined ex-
perimentally (for example, by measioring stresses in
the rotor shaft or in the reduction gear mount).
It should be noted that the qualitative methods
of estimating vibrations permits a number of useful
conclusions in designing helicopters and in iji^^rov-
ing them during flight tests. For exanple, it is
possible to judge the effect on vibrations of the
shape of the blade resonance diagram and the fuse-
lage resonance diagram and thus define the direction
toward which the design parameters should be changed
so as to reduce vibrations, and sometimes even to
estimate the degree of reduction in vibration.
Fig. 2. 2 Rotor Rotating
in an Oncoming Airflow.
To draw certain general conclusions as to the nature of time-variance of
the forces X(t), T(t), and Z(t) and of the moments Mx(t), My(t), and M^(t), let
us turn to Fig .2.2 which shows a 5-blade rotor uniformly rotating with an angu-
lar velocity uo in a relative airflow of constant velocity V. At a certain
time t, let the rotor blades occupy the position shown in the sketch and let at
this time the force X have a certain value X(t). After a time interval equal to
1/5 of the time of one conplete revolution of the rotor, the rotor will turn by
1/5 of this complete revolution. This causes blade No.l to occupy the position
of blade No. 2, blade No. 2 that of blade No. 3, and so on. It is obvious that, in
the new position and if all rotor blades are absolutely identical, the entire
pattern of flow and hence all forces acting on the blade will be exactly the
same as at the initial time t. In particular, the value of the force X will be
the same. It is evident that the situation is repeated with the next turn of
the rotor by 1/5 of a conplete revolution. Consequently, the function X(t) is a
periodic function of time of a period equal to 1/5 the time of one coirplete rotor
revolution. Figure 2.3 shows one of the possible slopes of the curve of the de-
pendence X = x(t).
Thus, the force X will vary in time with an angular frequency 5^, whereas
the variable forces acting on the rotor blade will change with a frequency uo
(once per rotor revolution).
215
Illlllllilllil
like any periodic function^ the function X(t) can "be e^q^anded in a Fourier
series. This will cause the lower harmonic in the expansion to be the har- /202
monic 5cd, so that the e^q^ansion will have the form
i.e., the fundamental frequency p = 5^, while the muJLtiple frequencies are 2p =
= lOcjD, 3p = 1503, Z^ = 20cju, etc.
Olbviously, exactly the same conclusions can "be drawn with respect to the
functions Y(t), Z(t), K^W) , My(t), andM,(t).
In general, for a rotor with a number of "blades equal to z, all forces and
moments acting on the helicopter periodically change in time, with the frequency
of the so-called fundamental harmonic of the rotor p = zcjd. The expansion of
these forces and moments in a Fourier series has the form
^ (0 = -^Q+-^a,cos pi + Xt,^ sinplJ^Xa,CQs2pi + ^
^jr W = ^^^a+^fl, cos pi + M^^sinpl+M^^ cos 2pi-{-
+ M^^sin2pti'M^^cos3pt+M^^slnSpi-^...;
(1.1)
where
p=z^. (1.2)
Thus, in a helicopter with a nimiber of "blades z, excitation of vi"brations
is possi"ble only with frequencies zoo, 2za), 3ztJo, etc. We note that this conclu-
sion is valid also for fluctuating aero-
dynamic forces generated by the flow repulsed
by the rotor and acting directly on the fuse-
/X-Xft) [ lage.
-One rotor revolution-
~ If, in addition, the helicopter is
equipped with an antitorque rotor having Za. r
blades and rotating with an angular velocity
uog^. p , then the fuselage will be acted ipon
Fig. 2.3 Possible Shape of the also by exciting forces containing harmonics
Dependence of the Longitudinal Pa.r = Za.r^a.r* ^a,r * 3Pa.r ^'^^'
Force on Time.
All these conclusions are valid only if
the rotor blades are perfectly identical. If
this condition is not met, low excitation frequencies u), 2^, 3^j etc. might ap-
pear. However, many e^q^erimental data - results of vibration and stress analy-
ses of structural fuselage members of various helicopters - show that the conr-
tent of lower harmonics is always so insignificant that they can be disregarded
in helicopter vibration analysis as well as in strength estimates of structioral
fuselage parts. This indicates that the state of the art in manufacture and the
demands imposed on the blade stiructure result in sufficiently small deviations
216
of individual blade quality.
We note that all above argtimenbs are fully applicable to investigations of
variable forces acting on the swaslplate of the automatic pitch control and pro-
duced by the rotor blades. Despite the fact that the moment of the forces act-
ing on the blade relative to the axial hinge (hinge moment) varies in time -with
a fundamental frequency oo, the resultant forces and moments acting on the swash-
plate vary in time with a fundamental frequency zu). Therefore, the variable
forces acting in the collective and cyclic pitch control loops vary with a /203
fundamental frequency p = zoo and also contain the harmonics 2pf 3p, kp, etc.
In addition, lower excitation hannonics can appear only at deviations in indivi-
dual blade properties.
Dep ende nce p_f t he P^equency Spectrum of Exciting Forces
on the Harmonic Content of Blade Vibrations
Above, on the basis of very general considerations, we have demonstrated
that variable forces and moments X, Y,
Fig. 2. 4 Polygon of Forces Generated
by the Blade and Inpressed on the
Rotor Hub.
Z, Mjc, My, and M^ produced by the vibra-
ting blades and acting on the rotor hub
vary in tijne with the frequency of the
fundamental harmonic zu) of the rotor and
also contain its multiple harmonics 2zu),
3zco, etc., whereas the rotor blades and
hence the forces generated by each blade
and acting on the hub perform vibrations
with the fundamental frequency au and
contain multiple harmonics 20), 3o), k^j
etc. which conprises also the harmonics
zcju, 2za), etc. This suggests that cer-
tain harmonic components of the vari-
able forces set up by each blade and ijn^
pressed on the hub are neutralized at
the hub while others are summ-ed. We
will prove that this is actually so.
Let us refer to Fig. 2. 4 which gives a
schematic sketch of a hub with hinged
blades .
The force iirpressed on the hub from the k-th blade can be resolved into
three conponents: \ directed along the blade radius, Pjc parallel to the axis of
the rotor shaft, and (^ perpendicular to both.
Each of these conponents is a periodic function of time with a fundamental
frequency uo. It is obvious that, in a steady-flight regime, the functions Ni^(t),
Pk(t), and (^(t) are identical for all blades but shifted in phase for each
blade relative to the adjacent one by some quantity correspoMing to the tame of
rotor turn through an angle 2tt/2. This justifies writing the expansion of these
functions in Fourier series in the form
^A = ^0 + ^«, cos (o)/ 4- <p^) + P,^ sin (o)/ + cp^) +
217
where
+ P„.cos2(«)^+<P^) + />*,sin2K+<P4)-|-...+
4-P„„COS«(a)/ + .p^) + P^^sin«(o)/ + <pjj)4-. . .^
<P*=— A (A=l,2, 3...Z),
(1.3)
or, more concisely^
In like manner, we have
n-I
(1.4)
(1.5)
(1.6)
Let us now formulate the following problem: Knowing the values of the co-
efficients of expansion in Fourier series of the functions Pic(t), Qic(t), and
^kC"*^)* or, in other words, knowing the hai^onic coirponents of the forces Pj^ ^ Qy.,
Njj, we find the variable forces X, Y, Z and the moments M^, My, M^ (more exactly
their harmonic conponents) from which we can plot the dependence of the vibra-
tion-inducing forces on various harmonic conponents of the forces produced by
an individual blade and acting on the hub#
Summing the forces generated by each blade and acting on the hub, we obtain
the following formulas:
ft-i
fe-i
where
^^ - azimuthal angle of the k-th blade:
(1.7)
(1.8)
(1.9)
(1.10)
218
2jt
^ft=*«^+9ft=**>^M — k\
(1-11)
h = distance between axis of rotation and matched flapping and drag
hinges.
If the hub has umnatched hinges, then we must take h = ly.h in eq.(l.9) and
h = l^.tj in eq.(l#lO)#
Let us examine in detail eq.(l«7) for determining the variable force Y.
Substituting into it'eq.(l«3) for the force V^, it becomes necessary to calcu-
late the sums in the form
2 cos n (oii + cpj^) and ^ sin ti (to^ -f ip^),
ft-i
A-I
where n are integers (n = 1> 2, 3, ...)•
¥e will show that the trigonometric sums of such a form have the following
noteworthy property: For any n not a multiple of the number of blades z, both /205
sums are equal to zero for any t; when n is a multiple of z, i.e., if n = sz
(s = 1, 2, 3, ...)> then
Yi cos SZ (u>^ -j- 9^) ^ z cos (szioi) ;
z
2 sin SZ (co/f -f 9j^) =:2: sin (sznyi).
(1.12)
For exanple, for a rotor with five blades (z = 5)
S cos ((i)/f-fcp^) = |] cos 2(0)2: +cp;^) =
5 6
= i;cos3(o)/ + (p^) =.^ tos4(o)^+cpj=0
ft-i
A-l
for any value of t, but
Furthermore,
2 cos 5 {ioi + cpit) = 5 cos (Sco^).
A-l
S 5
2 cos 6 (mt 4- cp^) == ^ cos 7 ((0/+ <p^) =
219
5 5
:=2 cos8(a)/+cpJ = 2cos9K+cp^)=0,
but 5
2cOSlO(a)^ + (pJ = 5cos(10(o/), etc.
We can prove the validity of eqs.(l.l2) by different methods. For this,
let us use the convenient method proposed by R.A.Mikheyev based on the applica-
tion of the well-known Euler formula expressing the relationship between trigono-
metric functions and exponential f-unctions with an imaginary argument. We will
prove the validity of only the first equation in the system (1.12). We have:
1 t.i
Therefore,
' -1 ft-i
---- in — k
« . 2« , z , 2tc.
in — k K-- —in — k
k^l A-I
Let us separately check the sum
2« ^ r . ^^
4^ in^^k f in?^\ I /„2!L\2 / -^ 2.N3 / 2.^
This is a geometric progression with the denominator e ^ . /206
Using the well-known formula for the sum of a geometric progression, we
obtain
Since n is an integer, the numerator of this expression is always equal to
zero, so that e^^"^^ = 1 (n = 1, 2, 3, •••)•
The denominator of this e^^ression can vanish only if ( \ is an integer,
i.e., if n is a multiple of the number of blades z. Thus, this siim is equal to
zero for any n with the exception of those n that are multiples of the number z*
In the latter case, the value of the sum becomes indeterminate (-;-)• This in-
220
determinacy can "be evaluated tjy the -well-known L' Hospital rule. lat n vary con-
tinuously, approaching some value sz (s is any integer; s = 1, 2, 3^ •••)• Dif-
ferentiating the numerator and denominator with respect to n and passing to the
limit n -♦ sz, we have
z , 2k .
in — h
E
€ ' =lim^ 'iim
n-t'SX n-*'Sz
• I2ne
I2%n
-=z.
(=^)
/2« —
We can also show exactly that
* _{n — ft f 0> if n is not a imiLtipIe of z;
Z, if /l = S2:,where 5= 1, 2, 3.
ft-1
As a result, we arrive at the conclusion that if n is not a multiple of z,
then
z
If n is a multiple of z (n = sz; s = 1, 2, 3^ • • •) $ then
z
yj cos /Itft = ~ {e^ "^' + ^- f n«() = ^ cQs rtu>/ = Z COS (SZini) .
In like manner, we can prove the validity of the second equation of the
system (1.12) .
form
The indicated property of trigonometric siims is conveniently written in the
_f f 0, if n is not a multiple of z;
2 cos n^f, =
A-i I z cos Alto/, if n==sz; 5=1, 2, 3...;
2 sin n^f^ = I
0, if n is not a multiple of z;
zsinnmi, if /t = sz; 5=1» 2, 3...
(1.13)
Let us now return to the expression for the force Y from eq.(l.7), into /207
which we substitute the value of the force Pj, from eq.(1.4):
>'=Il[^o + /'«.cos^,4-^*,sIn^, + P,.cos2>,+
+ Pft. sin 2^j^+ . . . + P^^cos n ^*+ /^a, sin n%+ . . .]
221
On the "basis of the established property of trigonometric sums [eq.(l«l3)]
it can be stated that, upon sijmmation of different harmonics in this expression,
all harmonics that are not a multiple of the niimber of blades 2 will disappear.
The harmonics that are a multiple of z are summed in conformity with eqs»(1.13),
so that we finally obtain
y=zPQ+zPa^coszi}it'{-zPf,^sinziiii^zP^ ^^ cos 2z(i)/ -[- zP^ sin2zu)^
(1-14)
Thus, all harmonic coirponents of the force P^ (t) that are not a multiple
of the number of blades are neutralized at the rotor hub and do not cause vibra-
tions of the helicopter fuselage. As a result, the variable force Y changes in
time with the fundamental harmonic p = zoo of
the rotor, and also contains multiple harmonics
a^^mm 2p, 3p^ etc. This conpletely confirms the
ftJ( — r^az — 1 — i r ' \"„u* 1 basic conclusion of the preceding Subsection,
and yields additional information exactly de-
fining the harmonic components of the force P^
that are dangerous from the aspect of vibra-
tions .
let us examine an exatiple for illustration
purposes. We assiome that, for some rotor, there
is resonance of the second overtone of blade
vibration in the flapping plane with the fifth
harmonic of the rotor (5tA)). In this case the
harmonic con5)onent corresponding to the fifth
harmonic (Pas and P^jg ) will be large in the ex-
pansion of the force F^ for such a rotor.
150 ykm/hr
Fig. 2* 5 Anplitude of Vibra-
tions in Cockpit of Single-
Rotor Helicopter as a Func-
tion of Flying Speed.
If the rotor has five blades, the above type of resonance will lead to ap-
preciable vibrations of the helicopter.
If the rotor has four blades, this resonance will in no way manifest itself
in vibrations of the helicopter, since the harmonic conponents of the force Pj^
corresponding to this resonance will be neutralized at the hub. As will be
shown later, the variable moments M^ and M^ at the hub can be appreciable; how-
ever, for all practical purposes the helicopter vibrations are determined mainly
by the variable forces X, Y, Z. Occasionally, it is erroneously assumed that
the vibrations of a given helicopter are smaller, the larger the number of rotor
blades. However, it is evident in this exanple that in reality the matter is
not so sinple and that in this case a reduction in the number of blades actually
will lead to a reduction in vibration.
Let us examine another exairple: Figiore 2.5 shows the results of eixperi-
mental vibration measurements in the cockpit of a single-rotor helicopter which
was tested with two rotors: three- and foior-blade types. The rotors had comr-
pletely identical blades and differed only in the hubs. The ciorves depict the
dependence of the anplitude ay of vertical vibrations in the cockpit on the fly-
ing speed V for both rotors.
As shown by calculations of these rotors, the rotor blade had a resonance /208
222
of the second overtone of vilDrations in the flapping plane with the fourth har-
monic of the rotor at the operating ipm. As a result, the vibrations of the
helicopter with a four-blade rotor, over the greater portion of the speed range,
were appreciably higher (at V = 40 - 50 km/hr, by a factor of more than 3) than
the vibrations of a helicopter with the three-blade rotor. However, at a high
flying speed the vibrations of the helicopter with a four-blade rotor were
smaller than those of the helicopter with a three-blade rotor* This is explained
by the fact that, at low flying speed, a large harmonic conponent of aerodynamic
forces exists, corresponding to the fourth harmonic and caused by the large non-
uniformity of the induced velocity field of the rotor at a low flying speed,
l&th an increase in flying speed there occurs an equalization of the velocity
field of the flow passing through the rotor (see Chapt.I, Sect .8); correspond-
ingly, the excitation of blade vibrations with respect to the fourth harmonic
decreases rapidly, whereas the third harmonic does not decrease as rapidly with
an increase in speed or may not decrease at all. The relatively large magnitude
of the fourth harmonic in the induced velocity field at low flying speed ap-
parently is a phenomenon common to all rotors.
Let us now return to a determination of other forces and moments acting on
the helicopter. Equation (1.9) for the moment My is coirpletely analogous to
eq.(l.7).
Repeating the reasoning used in deriving eq.(l.l^) for the force Y, we ob-
tain the following expression:
+ ^«(2^) cos2za)^ + Q,^^^^sin2z(o^+. , .]. (1-15)
The variable moment My is dangerous not only from the aspect of helicopter
vibrations (it will be shown in Sect.3> Subsect.l that this moment causes only
lateral fuselage vibrations). This moment is one of the sources of torsional
vibrations in the transmission system of a helicopter.
As we see from eq.(1.15), the variable portion of this moment is determined
exclusively by the harmonic conponents of the force Q(t) which are multiples to
the number of blades.
Let us now turn to the first equation of the system (1.8). Substituting
into it the expressions for Qj5:(t) and Nj^(t) [eqs.(1.5) and (1.6)], we obtain
-i; [A^a,costj>;^+A^,,sina>j,]cos6j,+...+2 [Q«^cos/i']^j,+
z
+ Qft„ sin n'^^] sin .j^ft - S {NaCos n%-\-N^^ sin «y cos <!'»+.. .
223
The first addend of this siom is equal to zero, in confonnity with eqs.(1.13)
since
To calculate the remaining addends let us examine the expression
z
which represents the conponent of the force Z caused by the n-th harmonic of /209
the force Q.
z z
Here, we encounter simis of the form S cos /i'^;^ sin 6;^ and 1] sin/^o^sin^j^ . These
sums are also easily calculated \>j means of eq«(l-13)« Actually,
z z z
2cos/i'^;,sinO^=-^ Jsm(/j+l)^ft \^ sin(/i-l)6^;
A-l ft-1 A-1
z z z
2sm/f;);,sin6j, = -^JJcos(a+l)'>^+^2cos(/i-ljO^.
*-i k~\ ft-i
On the basis of eqs-(l.l3) we can assert that these sioms will be nonzero
only if one of the numbers (n + 1) or (n - 1) is a multiple of the number of
blades* Let (n + 1) = sz (s = 1, 2, 3, •••) and thus n = sz - 1. Then,
z
V] COS rv^^ sin <^^ = — sin {sz^t)\
z
y] sin n^^ sin t;>^ = — ~ cos {sz^t).
Furthermore, if (n - 1) = sz; n = sz + 1, then
z
^ cos n% sin ^^^ = — -f. sin {sz^t)\
z
\\ sin n^f^ sin ^1^^=— cos {sz^t).
ft-i
A-:
As a result, we obtain the following e35)ression for the conponent of the
force X which is obtained from all harmonic conponents of the force Q:
22k
1 1 1 1 1 1 1 1 1 1
!■■■■■ !■■■■ ^m II iMHi I mill nil ii ii m
For the portion of the force X caused by harmonic components of the force
N(t), we obtain in like manner the e:j^ression of the form
(1.17)
The force X can be determined by the formula
If the e:xpression for the force X(t) is written in the form of eq.(l.l), /2IO
the following formulas for its harmonic components are obtained:
^''i— ^I*?i'(z+i)— Q*(;.-i)— ^«u+i)— ^«u-i)];
^^ ~ W« (.-1) ~ Q« u^i) - ^N.+u - ^^ (.-1)1 •
(1-18)
The coiiponents corresponding to harmonics that are multiples of the funda-
mental harmonic Xas^ ^ts* etc« are obtained from these same formulas, if we re-
place the index z by the indices 2z, 3z, etc.
Thus, the variable part of the force X(t) is determined by the harmonic
conponents of the forces Q(t) and N(t) which are combinatory with respect to the
fundamental harmonic of the rotor (z - 1; z + 1) or to its multiple harmonics
(2z - 1; 2z + 1), etc.
For exajiple, for a rotor with three blades (z = 3)* the fundamental har-
monic of the force X (frequency 3tDt) will be determined by the second and fourth
harmonics of the forces Q(t) and N(t), the second harmonic of the force (fre-
quency 6 out) will be determined by the fifth and seventh harmonics of forces Q(t)
and N(t), and so on.
Conpletely analogous formulas are obtained for harmonic components of the
force Z(t)
(1.19)
225
Just as in ec[s.(l«18), to olDtain the multiple hannoxiics Z^g, Z^g, Za3, Z^q
the index z in these formulas must "be suTDstituted respectively by the indices 2z,
32, etc.
In Hke manner, the expressions for the harmonic components of the moments
Mjj and M are otjtained from eqs.(l.lO):
(1-20)
^a^—Z^i — ^a (^4.1) ^ Pa (z^i)] ;
^^ = ^[~/^.<.^,)"/^.(.-u].
(1-21)
Let us also mention the following fact which occasionally might facilitate
a qualitative vi^bration analysis. If the varia*ble force in the rotor plane (X
or Z) or the moment (M^, M^) are determined hy some harmonic conponent of the
force generated 'by the blade, then we obtain a vector of constant length uni-
formly rotating in the plane of the rotor with an angular velocity zo) (or szod).
The direction of rotation is opposite to that of the rotor if this vector is ob-
tained from the harmonic conponent z + 1 (or sz + 1), and equidirectional with
the rotation of the rotor if this vector is obtained from the harmonic ccirponent
z - 1 (or sz - 1).
For instance, let the rotor have five blades (z = 5) and let us look at /2ll
the vector of the moment at the hub, with con^Donents M^ and M^ obtained from the
harmonic conponent (z - 1):
/^ -= /^a, cos 4^/^ + Pf,^ sin 4a)/.
Then [eqs.(1.20) and (1.21)],
zh
Mjc=--^ [Pb, cos b^yt - Pa, sin Sco/] ;
zh
M^=—[~PtMnb^t~~Pa,zos5^t].
As indicated by these formulas, the vector
represents a vector of constant length
M^'-^VPl^Ph
226
tmiformly rotating in the plane of the rotor with an angular velocity 5uj in a
direction coinciding with the direction of rotor rotation.
Thus, the above analysis shows that a rotor is a sort of filter which, out
of all harmonic conponents of the forces on vibrating blades, transmits to the
fuselage only certain ones corresponding to the fundamental harmonic of the
rotor zoo, to its conposite harmonics (z - 1) cd and (z + l)(jo, harmonics that are
multiples of the fundamental harmonic 2za}, 320), etc-, and to coirposite harmonics
(2z - l)uj, (2z + l)uo, (3z - l)cD, (3z + l)(JU, etc.
As a rtile, the lower harmonics zoo, (z + l)u), and (z - l)u) represent the
greatest danger both from the aspect of the vibration level and from the aspect
of dynamic strength of the fuselage members.
Of the harmonics which are a consequence of blade vibrations in the flap-
ping plane (force Pj^ , see Fig. 2.4), the harmonic zuj (and multiples of it) lead
to the appearance of a vertical variable force on the rotor, whereas the har-
monics (z - 1) and (z + l) (and also 2z - 1, 2z + 1, etc.) lead to the appear-
ance of variable moments at the hub relative to the axes Ox and Oz.
Of the harmonics which_are a consequence of blade vibrations in the plane
of rotation (forces ^ and Nj^, see Fig. 2*4)^ the harmonic zuo (and multiples of
it) lead to the appearance of a variable twisting moment on the rotor shaft,
whereas the harmonics (z - 1)(jd and (z + l)a), and also (2z - 1, 2z + 1, etc.),
lead to the appearance of variable forces (longitudinal and lateral) in the plane
of rotation of the rotor.
¥e note in conclusion that, upon summation of the forces generated by the
blades and acting on the swashplate of the pitch control, we obtain exactly the
same formulas for calculating the harmonic components of the vertical force Y and
the moments M^ and M^ applied to the swaslplate. In this case, eqs.(l.l4),
(1.20), and (1.2l) can be used directly, understanding by the force
Pk ii) = /'o + S {Pa, COS n% + P,^ sin «6,)
the force acting in the trimmer of the k-th blade (hinge moment divided by the
corresponding arm), and understanding by the quantity h the radius of the swash-
plate of the pitch control.
Thus, knowing the harmonic content of the hinge moment, it is not difficult
to calculate the variable forces acting in the collective and cyclic pitch con-
trol loops.
Section 2. Flex ura l Vibrations of the Fuselage as an Elastic Beam /2l2
If the variable forces in^arted to the fuselage by the rotors are known,
then calculation of \d_brations at different points of the fioselage can be car-
ried out by conventional methods of calculating the forced vibrations of an elas-
tic beam of variable cross section. Of course, the fuselage of a real helicopter
227
can "be regarded as a thin f lexurally elastic "beam.
In reality, the transverse dimensions of a fuselage cannot be considered
small in cognparison -with the longitiidinal dimensions. Purthermore, the fuselage
of a helicopter of single-rotor configuration may have "discontinuities" in the
region of the tail "boom, pronounced reduction in rigidity over the length, and
other peculiarities • These special features and their consideration in vibra-
tion analysis are discussed in Section 3* Here, we -will describe methods of
vibration analysis of an elastic beam, since these form the basis for further
discussion. In this Section, we will also investigate vibrations of a system
consisting of two elastic beams forming a "cross". A fiiselage with a wing is
reduced to such a system.
1. Calculation of Forced Vibrations of an Elastic Beam
by the Method of Expansion in Natural Modes
let a time- variant load q, distributed over the beam length and varying in
accordance with the harmonic law
q{x,t) = q{x)cQspt
(2-1)
be applied to a flexurally elastic ideal beam (Fig. 2. 6) without danping, which
is in a free state under the effect of a balanced system of time-invariant forces
(the force of rotor thrust balances the force of
gravity) .
Fig. 2. 6 Diagrain of a Free
Elastic Beam iinder Applica-
tion of a Distributed Load.
The equation of lateral f lexural vibra-
tions of such a beam has the form
{EIyy^my^q{x,t).
(2.2)
This equation in partial derivatives was
derived in Subsection 10, Section 1, Chapter I
for an elastic beam in a centrifugal force field.
In their absence (N t= o), the e:xpression takes
the form of eq.(2.2).
The problem is to find the motion of the
beajn, i.e., to find the ftinction y = y(x, t) which satisfies eq.(2.2) and the
boundary conditions which, in the case of a beam with free ends, have the form
at jc = 0; M = EIy"^0; Q = (£//)' = 0;
at jc=/; M = E/y" = 0; Q = {EIy'
(2.3)
The functions y(x, t) satisfying the homogeneous equation (without the
right-hand side)
^Elyy+my=^0 (2.4)
228
I I
and the "boundary conditions (2»3) correspond to the natiiral vibrations of the
beam. The solution of eq.(2«4) is sought in the form
y(x,i)=zy(x) cos pi.
(2.5)
This expression, after substitution into eq.(2.4), leads to an ordinary /213
differential equation with the parameter p for determining the fiinction y(x) :
{Ely')"- p^my=0.
(2.6)
The last equation has solutions different from zero only at certain values
of the parameter p: p = Po* P = Pi; P = P2 ; P = Pa ^ etc. To each value of p =
= Pk (k = 0, 1, 2, 3, •••) there corresponds a certain function yk(x), which
satisfies eq.(2.6) at p = p^ , so that
First fundamental mode n^Q
{Ely'i,)"-
p\my^=0; (4=1, 2, 3,.,.).
(2.7)
Second fundamental modeP~^
'Center of gravity
First elastic mode
Second elastic mode P-Pl
The orders of p^^ (k = 0, 1, 2, 3, ...)
are called the natirral frequencies of the
beam, while the functions J^i^) are desig-
nated as the corresponding natural vibration
modes .
law
The motion of the beam according to the
y(^,t)=af,y^{x)cospf,t,
(2.8)
Fig. 2. 7 Characteristic Natural
Vibration Modes of a Fuselage
as a Free Beam.
(p*!, ps, etc. are the vibration
frequencies of the first,
second, etc. elastic overtones;
in general we can assume: po =
= 0; pi = 0, P2 - p^l; P3 = pf,
etc.) •
where aj^ is a constant, is called the natu-
ral vibration of the beam with respect to
the k-th overtone.
The general solution of the homogeneous
equation (2.4) has the form
y{X, 0=S«*t/j,(JC)cOS (/?,/ + (?,), (2.9)
k
where a^^ and cp^ are arbitrary constants.
Thus, the natural vibrations of a beam
represent motion produced as a result of
the siperposition of vibrations of different
overtones .
The methods of finding the natural fre-
quencies pic and the corresponding modes 7k (^)
for a beam with a given law of variation in rigidity El(x) and a linear mass
m(x) are presented in Section 2 of Chapter I.
229
Plgure 2*7 shows the characteristic modes of natural -vibrations of a free
*beam« The two modes correspond to vibrations of a "beam as a solid body and. have
natural frequencies equal to zero. The first of these modes corresponds to for-
ward motions of the beam, and the second to angular displacement of the beam
relative to its center of gravity.
All formulas derived in this Section are equally suitable for calculating
the vibrations of an elastic beam with any clainping conditions at its ends .
However, when these formulas are used for vibrations of a free beam and particu-
larly of a fuselage, it must be remembered that the number of the frequencies Pi^
and of the modes ykCx) of natural vibrations must include the two lower modes
which correspond to fundamental frequencies. Thus, in all formulas it is neces-
sary to set Po = and Pi = and to take into account that the corresponding /2l4
nonned modes have the form
yi W
I — Xc
where Xc is the coordinate of the center of gravity of the beam.
If the above modes are not taken into account in calculations of fuselage
vibrations, the vibration analysis will not include vibrations of the fuselage
as a solid body, which will lead to appreciable errors in the vibration magni-
tude.
Let us study here the problem of forced vibrations of a beam subjected to
a "purely" harmonic load [see eq.(2«l)]. In this case, eq.(2.2) takes the form
{EIyy-\~my=q{x)cospL (2.10)
Let us first seek the particular solution of this equation corresponding to
steady forced vibrations of the beam with a frequency p in the form
y =^'y{x) cos pt. (2.11)
Substituting this expression into eq.(2.l0), we arrive at an ordinary dif-
ferential equation for determining the function y(x) which, of course, is known
as the mode of forced vibrations:
{El^y-p^nry = q{x). (2.12)
Let us then seek the solution of this equation in the form of an e^q^ansion
in nat-ural modes:
If, in this sum, we take a limted number of terms, then, in determining
the values of the coefficients Cj^, we can obtain only the approximate solutions
of eq.(2.l2). However, it is possible to prove that in the method of determinr-
230
ing the coefficients Cj^ given below, the approxiinate solution with a rather
large n-umtier of terms in the series (2#13) can differ from the exact solution
as much as desired.
To find the coefficients Ci^, we substitute eq.(2«13) into eq*(2«l2) and,
after multiplying both sides of eq.(2.l2) by yjj(x), we integrate them from
to I • This will yield the equation
I,cA{EIy\yy^dx~p^Y,cAmtj^y^dx^\q~y^dx. . .. .
The integrals in the first term on the left-hand side of this equation can
be sinplified by using integration by parts:
/ _, ,_ ^_ _, _ _, , ^ ^_,
f {Elyuyy^dx = f yj {Elyn) = [y^ {Elyu)'] ~ [ y'n {EIy\)dx,
6 6 " ^
but
\y,{Eryu)\
I
since the functions yic(x) satisfies the boundary conditions (2«3)«
Furthermore, /2l5
I y^niEIyiy dx^^ y,d{EIyu)=[yn{EIyk)] -^Elynykdx.
By virtue of the conditions (2.3)* we have
[^n(£/^;)]|^=-0'
so that, as a result, we obtain
J {ElylYy^dx = ^Elynyudx. (2.15)
Since all functions yic(x) (k = 1, 2, 3, ...) satisfy eq.(2.7)* we can write
{E/'ylY— plm7/f,=0\
{E/ynY— plmy„ = 0.
Multiplying the first equation by j^ and the second by j^ , we then sut^-
tract one from the other and integrate the obtained expression from to I .
This yields
231
i. I. I.
However, the left-hand side of this equation is equal to zero "by virtue of
the condition (2.15) • Therefore, if only pj, ^ p^, then
I
^ my^yf,dx=^Q; {ri^k\ (2.16)
This is the so-called condition of orthogonality of the natural vit)ration
modes (see also Chapt.I, Sect. 2, Sut)sect#3)*
Furthermore, multiplying "both sides of eq.(2.7) "by y^ and integrating from
to I, we obtain
J {E^ yk)" yndx= pi [ my^y^dx.
Hence, we can conclude that if n ^ k, then
[{EIy\Yy^dx^^EIyny\dx=Q. (2.1?)
If n = k, we obtain an e35)ression for the frequency Pn of the n-th overtone
of vibrations in terms of its mode j^ (x) :
J £h"ndx
Pl=—i • (2.18)
This is the well-known Rayleigh formula.
On the basis of conditions (2.16) and (2.1?) "we can assert that, in
eq.(2.14), all terms for which 'k ^ n vanish. Taking this into account and mak-
ing use of eq.(2.15), we rewrite eq.(2.1^) in the form /2l6
- -'2 I -9 ^
£?„ J Ely'n dx - c^p^ J my\dx= J qy^.
Dividing both sides of the last equation by J m^^dy:, solving it relative
to Cn, and using Rayleigh»s fonnula [eq.(2.18)], we find
232
^-==-
Igyndx
1
We then introduce the notations:
" ^--^^*j'_.-2.„' (2.19)
dx
The quantity A^, represents the work of the exciting load q(x) at the mode
of the n-th overtone of vibrations, while the quantity k^ denotes the largest
(during the period) value of the kinetic energy of the given overtone of vibra-
tions referred to the quantity p^ . Thus,
c = _i_^
" pI-p' Kn ' (2.22)
Taking into account eqs.(2»l3) and. (2«ll), we obtain the following solution
of equation (2.10):
^•'•''iSTrV-^I'.W
COS pi.
(2.23)
From this expression, we can draw certain iirportant conclusions.
First, it is obvious that if the frequency of variation of the exciting
load p approaches one of the frequencies p^ of natioral vibrations, then the vi-
bration anplitude at any point of the beam increases without bounds. This is
the phenomenon of resonance of an exciting load with the k-th overtone of natu-
ral blade vibrations. Since we do not consider here the effect of dairping forces
(this will be done later on), the vibration an^Dlitude in resonance is unlimited.
Furthermore, if the quantity p is close to the frequency Pn o^ "the n^th
overtone of vibrations, the term with the number n in the sum (2-23) becomes ap-
preciably larger than the other terms. Therefore, we can assume approximately
that, in the vicinity of resonance (p = p^), we have
meaning that, in the vicinity of resonance with some overtone of natioral vibra-
tions the mode of forced vibrations differs little from the mode of vibrations
of the given overtone.
233
yj -
Finally, when the value of p changes from an amount somewhat smaller than /2l7
Pjj to an amount somewhat larger than p^, the quantity in the brackets of formula
(2»23) changes sign. Therefore, if we construct a graph for the dependence of
the anplitude Jq of some point of the "beam
on the excitation frequency p [for a con^
stant q(x)], this graph will have the shape
shown in Pig.2.8# The curve of the graph
has infinite discontinuities at the points
P ^ Pi* P = P2> P = P3* etc.
2* Dynamic Rigidity of a Beam *
Resonance and Antiresonance
In the preceding SulDsection, we dis-
cussed the case of forced vLhrations of a
■beam sut>jected to an exciting force dis-
tributed over its length, which varies in
time iDj the harmomc law (2*1); the derived
formulas remain in force for any law of
variation in load over the heam length.
i.e#, for any form of the function q(x) •
Therefore, it is not difficult to derive,
from these e:xpresslons, formulas for deter-
mining the forced vihratlons of a heam
caused by a concentrated exciting force
Flg.2»8 Dependence of Vibration
Amplitude of any Fuselage Point
on the Excitation Frequency.
applied at a certain point x = Xq (Fig,2.9)«
(2.2^)
In fact, let the load q(x) be applied to a beam over only a small segment
of length Ax in the vicinity of the point x = Xq* In this case, eqs.(2.22),
(2.20), and (2.2l) remain valid, but in eq.(2.20) the corresponding integral
must not be taken over the entire length of the beam I but only over a segment
Ax,
x.ei
4jf
At a small value of Ax, this integral can be approximately replaced by the
quantity
where
j qyndx= Foy„{Xo),
4X
Fo = jqdx.
(2.25)
(2.26)
234
Equation (2.25) "becomes exact at an infinitely small Ax, i.e., in the case of a
concentrated exciting force.
Thus, we arrive at the following conclusions: If the vibrations of a /2l^
"beam are caused ty a concentrated force [eq.(2.2^)] appHed at the point x = x^,
then the motion of the l^eam is described as "before tiy eq.(2. 23) in which the
quantity Aj^ is determined by the formula
^k^^oy^Mf
(2.27)
i.e., the quantity Aj^ represents the work done by the exciting load "at the mode
of the k-th overtone of vibrations".
^F'FffCO&pt
\ /l X-X pcospt
Pig. 2. 9 For Analyzing Forced
Vibrations of a Free Beam due
to a Concentrated Force.
Fig. 2. 10 Diagram of the Action
of a Longitudinal Force Produced
by the Rotor and Exerted on an
Elastic Fuselage.
¥e note that this method of defining the forced vibrations holds also if
the vibrations are caused by a concentrated bending moment varying by a har-
monic law
M = MqCOs pt.
(2.2S)
appHed at the point x = Xq* In this case, the quantity A;^ should be determined
by the formula
^* = ^oi/U-^o).
(2.29)
where yj^'(xo) is the angle of rotation of the elastic line at the point x = Xq
corresponding to the mode of the k-th overtone.
If the beam vibrations are caused by a longitudinal force
X=^XoCOspt (2.30)
applied to some arm h (PLg.2.10), all of the derived formulas remain valid
since, in this case, the force Xq can be transferred from the point A to the
corresponding point B of the beam, during which process the coiple with a moment
equal to Mq = Xoh has been added.
235
/
The longitudinal variable force appHed at the point B is alDle to cause
only longitudinal (axial) vibrations of the iDeam, whereas lateral vibrations of
the beam due to the harmonic moment Mq are determined in the manner indicated
above.
In examining lateral forced vibrations of a beam produced by a concentrated
force F = Fq cos pt, it is convenient to introduce the concept of dynamic rigid-
ity of the beam at the point of application of the force x = Xo»
Let the dynamic rigidity D(p) of the beam at the point x = Xq be repre-
sented as the ratio of_the highest value (arrplitude) of the exciting force Fq
to the anpHtude Jq = y(xo) of the forced vibrations of the beam at the point of
application of force, such that
^(^)=f' (2.3X)
¥e have in mind that, on a variation in force in accordance with the har-
monic law F = Fq cos pt, the point of application of this force will execute
steady forced vibrations according to the law y = yo cos pt.
Thus," the dynajnic rigidity of a beam is a function of the vibration fre-
quency p and is considered positive if the force and displacement vary in time
"in phase" and negative if the force and displacement vary in "antiphase".
The vibration anplitude of the point of application of the force x = Xq /2l9
can be determined from eq.(2.23)*
yo
-S^V^»'<^'
pI^p^ Ku'^'-' ^^' (2-32)
If we plot th^ graph of the variation of yo with respect to the frequency p
at a constant value of Fq, a curve analogous to that shown in Fig. 2. 8 will be
obtained. Therefore, if we construct the graph of the dependence of the dynamic
rigidity D(p) at the given point of the beam as a function of the vibration fre-
quency, this graph will have the form shown in Fig. 2. 11.
The dynamic rigidity D(p) vanishes at the resonances p = Pi, p = Ps^ etc.
and becomes infinite at all values of the frequency p (p = P12* P "^ P33* P ~
^ P34> etc.) at which the vibration amplitude of the point of application of
force vanishes- These values of the frequency p are known as antiresonance fre-
quencies and are equal to the frequencies of the corresponding overtones of
natxiral vibrations of the beam with a hinged support at the point of application
of force F.
Actually, let us imagine that at the point of application of force F the
beam has a hinged sigpport (the beam is not crosscut at this point) so that this
point of the beam remains stationary during vibration. Such a beam has its own
natural vibration frequencies and modes. In the presence of natiiral beam vibra-
236
B{^)kg/cm
p 1/sec
tions of a certain overtone, a dynamic re-
action will arise at the. support x = Xq
which varies in time according to a har-
monic law with the frequency of this over-
tone. The ajiplitude (highest value) of
this reaction force will depend on the
anpHtude (of some point, for exanple, the
end) of nat\jral vibrations of the beam,
which may have any magnitude (depending
upon the initial conditions)* Therefore,
we can always select a beam vibration am-
plitude such that the reaction force anpli-
tude has a prescribed value Fq . If we now
imagine the st5)port as removed but still
continue to apply, to the beam at this
point, the force F varying by a harmonic
law with the same frequency, then the free
beam will continue to vibrate with respect
to the same mode with the same amplitude*
However, these vibrations can be regarded
as forced vibrations of a free beam under
the effect of the exciting force F. With
such forced vibrations, the point of application of the exciting force is sta-
tionary so that the dynamic rigidity of the beam, corresponding to this regime
is infinite. This is known as antiresonance*
Fig. 2. 11 Graph of Dynamic
Rigidity.
In the graph of the dynamic rigidity (Fig. 2. 11), the points of resonance
D(p) - and antiresonance D(p) = oo alternate. It can be demonstrated that this
is always so for an elastic beam.
Thus, at a certain excitation frequency, the point of application of the
exciting force becomes arrested, and the node of the forced vibration will be
formed at this point. This phenomenon is called antiresonance. The frequency
of each antiresonance is always located between two adjacent natural vibration
frequencies of a free beam.
The phenomenon of antiresonance in "piore form" can occur only in ideal /220
oscillatory systems without damping. In the presence of danping, the vibration
anplitude of the point of application of the force in antiresonance does not
vanish. This anplitude will be lower, the smaller the danping [see, for ex-
airple, the paper by Den-Gartog (Ref.l9) on a dynamic vibration daiiper) .
3 . Application o f t he^ Method, of Dynajni c Rigidity to the Vibration
Analysis of Side-by-Side Helicopters
The concept of dynamic rigidity is rather convenient in calculating oscil-
latory systems that can be divided into two or more components, making it easy
to define their vibrations individually.
let us examine a vibratory system consisting of two crossed elastic beams 1
and 2, shown in Fig.2«l2. A fuselage with an elastic wing, characteristic for
237
helicopters of side-"by-side configuration, represents such a system.
It is necessary to calculate the forced vibrations of this system caused by
a variable force F, varying according to a harmonic law and appHed at the
coT5)ling point A of the beams 1 and 2 (the
method of calculation will be indicated be-
low for the case in which the exciting
forces are applied at any point). Using the
method presented in Subsections 1 and 2, it
is possible to calculate, for each of the
beams, the forced -vibrations produced by
certain forces Fi and Fg applied to each of
these beamg at the point A. In so doing,
we can find the dynamic rigidity of each of
the beams at the point A. Let these dynamic
Fig.2«l2 Diagram of Vibratory rigidities be D^Cp) and Dg(p).
System of Two Crossed Beams.
It is easy to show that the dynamic
rigidity D(p) of the entire system will be
equal to the sum of the dynamic rigidities of both beams:
Z)(p)=Z)i(p)+i)2(p).
(2.33)
Actually, the force F = Fq cos pt acting on the system as a whole will be
equal to the sum of the forces F^
each of the beams. However,
Fqi cos pt and
Fs =
Fq2 cos pt acting on
^oi=^A(/?)^o;
^02==A(/^)^0>
where Jq is the vibration anpHtude of the point A, identical for both beams.
Consequently,
^o-=^oi+/='o2=[A (P) + D,(p)] l, = D (pfy,.
Thus, the dynamic rigidity of the system is easily found by means of
eq.(2.33) if the dynamic rigidities of the beams 1 and 2 are known. The graph
of the dynamic rigidity D(p) can be obtained by simple addition of the ordinates
of the graphs .Dx(p) and B3(p). The values of the frequency p at which D(p) =
will give the values of the natural frequencies of the system of two beams. This
yields a convenient method for determining the natural frequencies of the sys-
tem. Since these frequencies are the roots of the equation
D(p)=Z)i(p)-fD2(p)=0,
they can be found from the condition
D:{p)^-D2ip),
/221
(2.34)
238
The last equation is easy to solve graphically ty supeiposition of the
graphs of Di(p) and -D^Cp), as is shown in Eig.2»13» The abscissas p^, pg, etc#
of the points of intersection of the graphs D^Cp) and -DgCp) give the values of
the natiiral frequencies of the system.
mp)
With this method of calculation,
the natural vibration modes of the sys-
tem are simultaneously determined. The
natural vibration mode of the system,
corresponding to some frequency p^
(k = 1, 2, •••), will consist of the
forced vibration modes of each of the
beams at this frequency, due to the
■02 "
Fig.2»l3 For Determining the
Natural Frequencies of the System
by the Method of Dynamic Rigidi-
ties •
forces Fqi and F(
frequencies,
it follows that
^01= /^02.
Since, at natural
= 0,
i.e., the force F^-^ applied to the
beam 1 is equal in magnitude and op-
posite in sign to the force Fog applied
to the beam 2*
The natural vibration modes of this system can be normed by selecting an
appropriate scale. For exaiiple, it is possible to select a scale such that the
vibration mode of the beam 1 has an anplitude equal to unity at its tip (x = t).
In this case, the corresponding scale of the vibration mode of the beam 2 should
be selected from the condition of a vibration anplitude identical with the
beam 1 at the coupling point.
Having the normed natural vibration modes of the system available, its
forced vibrations can be calculated from harmonic forces applied at any point,
by the method of expansion in normal modes in the same manner as in the case of
an isolated beam. Here, the vibrations of both beams are sought in the form
y{x, t)=^^c^y^{x),
(2.35)
where yic(x) is the vibration mode of a given beam corresponding to the normed
mode of the k-th overtone of vibrations of the system (simultaneous vibrations
of both beams) .
The coefficients c^ are determined in the conventional manner from eq.(2.22):
_ 1 Ak
(2.36)
where pj^ is the frequency of simultaneous vibrations of the k-th overtone of the
system.
239
las:
The coefficients Aj^ and K^ are determined "by means of the following formu-
Ak^^oyuM' (^-1,2, 3....)
(2.37)
This coefficient represents the work done "by the exciting load at the mode
of the k-th overtone of natural vibrations of the system. The quantity yic(xo)
represents the anplitude of the normed vilDration mode of the k-th overtone of
the system at the point of application of force, .regardless to which "beam the
excitation is appHed [here, yk(xo) is taken with a "plus" sign if the direction
of the force and the deflection coincide, and with a "minus" sign if the di- /22:
rection of the force and deflection do not coincide]:
for 1st beam for 2nd beam
(2.38)
If vi"brations of the system
are excited ^^J several harmonic
forces appHed to different points
instead of "by a single force, then
the forced vi"brations are found "by
adding the vibrations caused by each
of the forces separately.
Here, we should briefly men-
tion one of the peculiarities of ex-
citation by rotors of helicopters
of multirotor configuration. De-
pending on the kinematic connection
of the rotors (over the transmis-
sion system), it may happen that
variable exciting forces produced by
different rotors vary in time in
phase or in antiphase. For exairple,
if the rotors of a side-by-side
helicopter are so coipled that the
blades of both rotors simultaneous-
ly occupy analogous positions (for
exanple, extreme forward position
as shown in the diagram A of
Fig.2.1fi-), the forces exerted on
both rotors simultaneously attain
•maximum and minimum values - they
will vary in phase. If the rotors
are coupled as shown in the diagram B, then- the exciting loads from both rotors
vary in antiphase. In case A, the exciting loads from both rotors will cause
only symmetric modes of simultaneous vibrations of the fuselage-wing system
whereas, in case B, only skew-symmetric modes occur (Fig. 2. 15). Since, in the
case of skew-symmetric vibrations, there are no vertical vibrations of the fuse-
lage points for helicopters of side-by-side configuration, it is desirable to
Fig. 2. 14 For Analysis of Vibrations
of Side-by-Side Helicopter.
240
SkeW' symmetric
modes
Symnieiric
modes
Fig,2#15 Natural Vibration Modes
of a Wing-Fuselage System in a
Side- by-Side Helicopter #
connect the rotors as shown in the dia-
gram B (Fig. 2.1^). Analogous considera-
tion can be made with respect to tandem
helicopters.
Of course, in solving the problem of
the most suitable mutual arrangement of /223
rotors it is also necessary to consider
the specific values of natural frequen-
cies of various overtones of the fuselage
and to examine, along with fuselage vi-
brations in the plane of symmetry, later-
al vibrations; this will be discussed
further in Section 3»
4* Metho d of Auxiliary Mass
To determine the dynamic stiffness by the method proposed in Subsection 2,
results from a natural vibration analysis of the fuselage are required. In this
case, the aiiplitude of forced vibration of the point of application of force^
needed for determining the dynamic stiffness, is determined by means of eq.(2.32)
as an e:xpansion in natural modes. However, whenever it is possible to program
the calculation of natural frequencies on a digital coxrputer so that this calcu-
lation will take little time, we can recommend the so-called method of aioxiliary
mass for determining the dynamic stiffness of the fuselage at a given point. In
this method, the natural fuselage frequency is calculated under attachment of
an auxiliary mass Am to the point at which the dynamic stiffness is to be deter-
mined. The calculation is performed for different values of Am, and its results
are used for plotting the graph Am(p) of the dependence of Am on the natural
frequencies of different overtones.
AG kg
10000
I
!
1
;i
\
1 ii
1-
-- '
1
! i
1
1
""
■r
■ 1
-'
\
i ii
ii
\
1
1
1
\
!i
i~
-■
I:
mo
1
1
1
1
Iv
'l00QY-n03\
^fsoo-
r-woo^\
pcyclmm\
\
3 c 6
O u.
- a L.
CJ
\
C 6
IS-
i
-CO""
^ o."
— h
—
-L-^
:"i
[
Fig. 2. 16 Typical Dependence of AtocLliary Mass of an Elastic
Fuselage (or Dynamic Stiff ness) at the Point of Rotor Attach-
ment on the Excitation Frequency.
241
Figure 2.16 gives an exanple of such a graph for a single-rotor helicopter.
In this diagram, the weight of the extra mass AG = gAmis laid off on the ordi-
nate.
It is easy to show that this graph, to some degree, can conpletely replace
the graph D(p) in Fig. 2.11- In fact, for natural vibrations of a tieam with an
auxiliary mass Am at a frequency p, the "beam will "be loaded "by the correspond-
ing additional force of inertia whose anpHtude is
Fo=Am/?2tfo, (2.39)
where Jq is the vibration anpHtude at the point of attachment of the auxiliary
mass.
The force of inertia Fq at the instant of maximum deflection from the /22^
equilibriim position is directed toward the same side as the deflection Jq» A
spring attached to a beam with a stiffness |c| = |Aiip^|, producing a force pro-
portional to the deflection y^ and directed opposite to this deflection, cor-
responds to negative values of Am.
Of course, exactly the same vibrations of the beam can be obtained without
an auxiliary mass, but these are forced vibrations produced by the action of a
harmonic force of the same amplitude Fq and vai^ying with the same frequency p.
The dynajnic stiffness of the beam is determined by means of the formula
yo
Comparing this e^^ression with eq.(2.39)^ we find
D{p)==p2^tn(p). (2.40)
On the basis of this formula, it is easy to construct the graph of the de-
pendence D(p), since we have the dependence Am(p) at our disposal. However,
this need not be done and the graph Am(p) or AG-(p) can be used directly. For
exanple, to determine the natural fuselage frequencies of a side-by-side conr-
figuration, we can locate the point of intersection of the graphs AGi(p) and
-AGgCp) instead of the points of intersection on the graph D^Cp) and -D^Cp) (see
RLg.2.13)-
5. Effect of Damping Forces. Vibrations at Resonance
The theory presented above and the resultant methods of calculation are
based on the assimption that the beam is perfectly elastic and that danping
forces are absent. Aq for ariy other oscillatory system, a vibration analysis of
a beam far from resonance need not take the danping forces into account; this-
does not lead to large errors.
However, a vibration analysis of a beam close to resonance or actually in
242
resonance requires allowance for the danping forces, since the vibration anpli-
tude at resonance is determined exclusively "by the presence of danping and
since, if absence of damping is ass^umed, the anplitude at resonance becomes tu>-
bounded •
Danping forces during vibrations of an elastic beam are generated mainly as
a consequence of friction between structural elements of the beam during its de-
formations and also as a consequence of so-called internal friction in the beam
material which, for a composite beam, is generally negligible in conparison with
the friction between structural elements (Ref •16).
The equation of f lexural vibrations of a beam in the presence of danping
can be derived by assuming that the bending moment M in the beam section is pro-
portional to its curvature ^ (in accordance with Hooke's law) and to the
time rate of change of curvatiure, so that we can write
+ iS-(^'S)' (2-W)
where T] is some coefficient characterizing the danping properties of the beam
at a given cross section, which is assumed to be a given function of the x-coor-
dinate.
Using the known relationship: /225
where q'^Cx, t) is the intensity of the lateral load applied to the beam, and
taking into account that this load, during vibration, is conposed of the ex-
ternal exciting load q(x, t) and the load due to inertia forces, so that
then eq.(2.4l) will yield the following partial differential equation describing
lateral vibrations of a beam with danping:
£(^'S)+''^(^'S)+-&'-'<-"- (2.4a)
This equation differs from eq.(2.2) only by the presence of a term with a
factor H; if Tl = 0, it will coincide with eq.(2.2).
If q(x, t) - 0, we obtain an equation describing the natural vibration of
a beam in the presence of danping:
2k3
U"B]+^^('^'ff:)+-B=<'- (2.43)
The exact solution of this equation is rather conplex. However, at rela-
tively weak dauping, a sinple approximate solution can "be used. Such an approx-
imate solution of this equation, corresponding to natijral vibrations of a iDeam
with respect to the k-th overtone, can "be found by assuming
where yic(x) is the natural vibration mode of the k-th overtone of the beam in
the absence of dairping#
Substituting this solution into eq.(2«43)^ canceling the factor e^k*, mul-
tiplying by yic(x;, integrating the equation within the interval to I , and
taking eqs.(2«l7) and (2*18) into account, we obtain the following equation for
determining Xy. :
ll + 2nf,l^+pl=^0, (2.45)
where ^
^n,=^'^^riEiy,'dx (2.46)
The roots of this equation will be
K=--^u±^Pl^ (2.47)
where
pI-VpI-^I' (2.4S)
Accordingly, we can write eq.(2.4^) in the form
y=l/^ (x) e-'^' cos {plt+ o), ( 2.49 )
i.e., the quantity nj^ represents the darrping coefficient of vibrations of the
k-th overtone while p^^ represents the natural frequency of the k-th overtone in
the presence of danping.
We can show that such an approximate solution of eq.(2.43) will differ /226
less from the exact solution the smaller - in comparison with unity - the dimen-
sionless coefficient of darrping of the k-th overtone determinable by the formula
This coefficient is one of the most inportant characteristics of vibrations
of the given overtone and can be determined esperitnentally, either by analysis
of the oscillogram of danped vibrations of the given overtone or by applying the
3W-
results of meastiring the forced vibration atiplitude of the "beam under the effect
of a vibrator (to be discussed below).
For a conventional fuselage (riveted fuselage with duralumin skin) the
danping coefficients n^ of different overtones are located v/ithin limits of 0.02
to 0.05 • These are rather small values of the danping coefficient, in whose
presence the vibration frequency of the k-th overtone can be considered equal
to the frequency calculated without consideration of danping, since p"^" =
= Pic yi - nif . This correction is insignificant for the indicated values of n^*
In calculating forced vibrations of a beam with danping, described by
eq.(2»42), it is preferable - in. view of the weak danping - to use an approx-
imate method based on the fact that danping is conpletely disregarded far from
resonance whereas, close to resonance, an approximate solution is obtained on
the assunption that the natural vibration mode near resonance of the k-th over-
tone, just as in the case of absence of danping, is close to the natural vibra-
tion mode of the given overtone.
In the presence of danping, the equation of forced vibrations of a beam
under the effect of a harmonic load
g{x,t)=g{x) cos pi (2. 51)
is conveniently written in the coftplex form
dx-i\ dx2j ' ^dx'idt \ dxV ' a^2 ^^ ^ (.2.52;
Since the real part of the right-hand side of this equation coincides with
eq.(2«5l), the actual motion of the beam is described, in view of the Unearity
of the solution, by the real part of the conplex solution of eq.(2.52). Close
to resonance with the k-th overtone of natural vibrations, the solution of this
equation in conformity with the above considerations is best sought in the form
y{xj) = c^y^{x)e^p^, (2.53)
where yfc(x), as usual, is the mode of the k-th overtone of vibrations in the ab-
sence of danping.
let us substitute this expression into eq.(2.52). We then multiply both
sides of eq.(2#52) by yic (x) and integrate from to I . Transforming the ob-
tained integrals and taking eqs.(2#l7) and (2»1S) as well as eq.(2»46) into ac-
count, we obtain the following equation for determining the coefficient Cij. :
where Ajj and K^ are as T:isual determined from e<^.(2.20) and (2.21). Hence,
2k5
The modulus of the conplex quantity Ci^ determines the viTbration amplitude ; /227
while the argument c^
argc,= A2^-^f_?^\ (2.56)
\ p^—pi I
determines the phase of the forced vibrations with respect to the exciting load
[eq.(2.5l)]. In the presence of resonance, the value of Cj^ [see eq.(2»54T] tie-
comes purely imaginary:
This means that, at resonance, the phase angle between the exciting load
and the vilDrations of the "beam is equal to tt/2. In this case, as is readily
verified by direct substitution into the equation, the vibrations will take place
in accordance with the law
f/(x,^)=7^^^(x)sin/7/, (2.57)
where
c^ = -
^n,PlKk '
{2.5B)
Thus, the vibration amplitude at resonance is conpletely determined by the
value n^ of the dimensionless coefficient of danping of the k-th overtone. This
can be used for an e:xperimental determination of n^ . If vibrations of the beam
are excited by means of a vibrator, i.e., by a given concentrated force F = F^
cos pt applied at a certain point x - Xq, and if the vibration anplitude Jq at
resonance (p = p^) is measured at the point of application of force, it becomes
easy to find the quantity nj^ . Here k^ will be determined by eq.(2.27) and the
quantity yo, by the for^nula
Therefore, taking account of eq.(2.58)f we find
1 Po[yk{^^)Y
n
'* 2 mp\Ku
or
1
2 rr^kPly^
^^^ 9 r..nK. - (2.59)
where the quantity 111^, which we can call the mass of the k-th overtone reduced
to the point x = Xq, is determined "by the formula
mf, = ^myldx.
Here,
Uk
(2.60)
(2.61)
The value of the reduced mass m^ is determined with sufficient accijracy "by
calculation, but it can also be determined experimentally "by measuring the natu-
ral vibration mode of the beam at resonance with the k-th overtone.
When desiring to make a pre-estimate of the anplitude at resonance for a /228
fuselage still on the drawing board and not yet given over to manufacture, it
is possible to use eq.(2.5S), using for n^ the values known from some other fuse-
lage of similar design, since the values of n^^ ^or sijnilar designs differ little.
Section 3» Vibration Analysis with Consideration of Fuselage
Characteristics
1# Fuselage Charac teristic s. lateral and Vertical Vibrations
In the preceding Section, methods were proposed for calculating the vi-
brations of a fuselage as an elastic beam (or as a system of two crossed beams
for a side-by-side configuration)
for which the dimensions of the
cross sections were small in com-
parison with the length. In many
cases, such a method of calculation
gives conpletely satisfactory re-
sults . However, in some cases
when the fuselage of the helicopter
has characteristics that differ
greatly from those of the model of
an elastic beam, more complicated
calculation systems are involved.
The fuselage designs of various
types of helicopters (single-rotor,
side-by-side configurations, tandem
configuration) vary widely. There-
fore, it would be difficult to give any generally applicable method of calcular-
tion which would permit a sufficiently accurate analysis of fuselage vibrations
generated by certain forces.
Each new fuselage design may necessitate substantial changes in the method
of calculation of vibrations. This problem might become rather conplicated.
However, in all cases the method of calculation should be based on general prinr-
ciples of the theory of vibrations of elastic systems. The design engineer who
Flexural axis
Fig. 2. 17 For Reducing the Vibration
Problem of an Elastic Fuselage to the
Vibration Problem of an Elastic Beam.
2kl
has the fimction of making vibration analyses of new configiirations for hell-
copter prototypes should be so versed in these general methods as to be able to
modify each conputational system to fit each new problem. Therefore, the ma-
terial in this Chapter is presented in a manner to demonstrate the essence of
the most inportant methods used in vibration analysis. For exairple, the method
of e^^ansion in natural modes, the method of dynamic rigidity, the concept of
resonance and antiresonance are not only applicable to an elastic beam or to a
system of two crossed beams but also to any other more conplicated vibratory sys-
tem. These methods were presented in their application to a beam since, on the
one hand, it is easiest to demonstrate them for this exanple and, on the other
hand, the method of calculating vibrations of a beam is often applicable to
fuselage vibration analyses without modification.
To illustrate certain characteristics of a real fuselage, let us turn to
I^g.2«l7 which schematically shows the fuselage of a single-rotor helicopter.
This fuselage is characterized by the fact that its flexural axis is a broken
line, that the centers of gravity of the fuselage conpartments do not lie on the
flexural axis, and that each fuselage coirpartment is a body all of whose measure-
ments are of the same order so that, in calculating vibrations, not only the /229
mass of the conpartment but also its
moments of inertia relative to all
J^y ly'^'^ y three axes must be taken into consid-
Ok
eration. Calculations show that, in
rpi determining the lower harmonic of
Q flexural vibrations of such a fuselage
^ n ^ i both in the plane :i£ij (vertical vibra-
"" tions) and in the plane xDz (lateral
2* z^/ j^ ^ vibrations), we can obtain coirpletely
^ satisfactoiy results if we conceive
^*^'^^^ the fuselage as a thin elastic beam
with a rectilinear axis.
Fig. 2. 18 Design Model for Vibration
Analysis of an Elastic Fuselage. If, in the vibration analysis, we
limit o-urselves to a study of vibra-
tions of the fuselage as a solid body
and take into account only the lower elastic mode (the first three modes in
Flg.2#7), the calculation of vibrations of a fuselage as a thin beam with a rec-
tilinear axis gives satisfactory results. However, if the second elastic mode
has a frequency close to the frequency of the fundamental harmonic of the
rotor zu) (and this is often the case), this type of calculation may lead to cer-
tain errors. In vibration analyses of the cockpit (at the fuselage nose) the
error may be insignificant while the vibration ajiplitudes in the region of the
tail boom may differ greatly from the real values. To increase the accuracy of
the calculations the vibrations must be determined with consideration of a large
n-umber of elastic overtones (second and third). However, a sufficiently accu-
rate determination of the second elastic mode now involves a conplication of the
calculation model.
An appreciable refinement of the calculated results can be obtained by using
the design model shown in Fig.2.1fi. The fuselage here is replaced by an elastic
beam with a rectilinear axis, to which individual loads 1, 2, 3> etc. are at-
tached. The center of gravity of each load is at a certain distance 1:^ from
2kB
the "beam axis. For each load, we assign its mass mi^ and moments of inertia I^
and I55 with respect to axes parallel to the axes Ox and Oz and passing through
the center of gravity of the load. For each segment of the elastic "beam between
the loads k and k + 1, we prescrilDe the flexural rigidities EI^ and KEJ in both
planes xOz and xDy and the torsional rigidity G\ •
z c^
J
Fu
ndamental\ pg^Q
^^
JPo
-J=^
.^■^r:
—
X m
Z if
5i5
Second ovtrion^^ pl^kkQcycfmin
/
Z (jD
Thi.
-d overtone'^ p^ =S3lCyc/miff
Q
^"--^
.^cr
^
^_
\
N
>J
X m
1
-5
5
Fig. 2. 19 Natural Vibration Modes of an Elastic Fuselage
of a Single- Rot or Helicopter in the Plane of Symmetry.
For this design model, the lateral vibrations (in the plane xOz) represent
simultaneous flexural and torsional (binary) vibrations. The frequencies and
modes of the natioral binary vibrations of such a system can be calculated by the
method proposed in Section 6 of Chapter I (see Fig.l.19) as applied to a rotor
blade. In this case, it must be assumed that the centrifiogal force N = 0, that
the rigidity of the control lines Ceon ^ 0, as well as that EIy''(o) = and
(EI^O' ^_q • This corresponds to the fact that the left end of the beam is not
clanped. The quantity x^.g in the blade calculation must be substituted by the
values of staggers hjj.
In calculating the forced lateral vibrations of this system, the method of
e^^ansion in natural modes (binary) can be used. In this case, all formulas of
Section 2 of this Chapter are applicable in which the quantity ky. means the work
done by the exciting load at the normed mode of a given haimionic and the quan-
tity Kj^ represents the kinetic energy of a given harmonic referred to the square
of its frequency pf . Figxare 2.19 shows the characteristic modes of natural, /230
lateral binary vibrations of a single-rotor helicopter.
The calculation method and model given in Fig. 2. IS can be used for an analy-
sis of vertical binary vibrations of the wing of a side-by-side helicopter with
wing-tip engine pods (Fig. 2. 20). If the centers of gravity of the pods have a
large offset h, the vibration analysis of such a wing cannot take only isolated
flexural vibrations in a vertical plane into consideration but must allow also
for simultaneous binary vibrations. A calculation of synchronous vibrations of
249
the fuselage-wing system in this case requires the method of dynamic stiffness.
The design model best simulating an actual helicopter fuselage obviously is
that shown in Fig.2»2l« Here, the flexxaral axis of the beam is given as a cer-
tain discontinuous line. The angle of inclination of the k-th segment of this
offset line is denoted as the angle o^i^ • Such a design model satisfactorily re-
flects the properties of any fuselage having a plane of symmetry xOy. For a
fuselage with such a plane of synmetry, a separate calculation can be made of
the vertical flexural vibrations (or vibrations in the plane of symmetry) and
the lateral binary vibrations.
Fig. 2*20 Diagram of Engine Pod
with large Offset.
Fig. 2.21 Design Model for Calculating
Vibrations of an Elastic Fuselage
with a Discontinuous Flexural Axis.
In calculating the vertical vibrations for each load, three degrees of
freedom must be taken into consideration:
displacement of the center of gravity of the load along the axis Ox;
displacement of the center of gravity of the load along the axis Oy;
rotation of the load relative to the axis Oz.
Z221
In calculating the lateral binary vibrations for each load, three degrees
of freedom must again be taken into account:
displacement of the center of gravity of the load along the axis Oz;
rotation about the axis Ox;
rotation about the axis Oy.
Calculation of vertical vibrations of such a system is discussed in the
Subsection below. We will also illustrate application of the so-called methods
of residues for vibration analysis, which often is rather convenient to use.
The calculation of lateral vibrations of such a system is not discussed
here since, for calculating lateral binary vibrations, rather satisfactory re-
sults can be obtained by using the design model shown in Fig. 2. IS. It should be
noted that, for the system shown in Fig.2.2l, the calculation of lateral binary
vibrations could also be carried out by the method of residues.
250
2» Calculation of Fuselage Vibrations in the Plane of Synmetry
"by the Method of Residues
Let a two-dimensional elastic system, depicted in Fig,2«2l, execute steady-
forced vibrations in its own plane xoy under the effect of a harmonic exciting
load consisting of forces and moments
Pky = Pli/ cos pi;
(3.1)
applied to each load (Fig#2»22).
During steady vibration, all points of the system will execute hamonic
vibrations with an excitation frequency p so that, if we denote by x, y, and d^
respectively the displacements of the center of gravity c of the load along the
axes Ox and Oy and the angle of rotation of the load relative to its center of
graArity, then we can express the k-th load by
X^^^Xj^COS pt\
y=-'ykCospt;
^=^f,cos pt
(>fe=l,2,3,
(3.2)
Let us then establish the relations connecting the forces applied to the
loads with the deformation of the beam segments. We will consider the forces
and deformations only for the position of the system corresponding to the maxi-
mum deviation from the position of equilibri-um (i.e., we will study only anpli-
tudes of forces and deformations). We then construct the equations of equilib-
rium for the k-th load (Fig. 2. 23) • To the load, the following are applied:
external forces P^x * Pky > ^k (applied at point A);
inertia forces of the load mjjp^Xjc; ^TgP^y^^l ^kP^^k (applied at point C);
forces acting on the load from the segment of the beam to the left of
it: \~x, ^k-i* ^-1 >
forces acting on the load from the beam segment to the right of it: /232
^k * Yfc * ^k •
The equations of equilibrium of the load are written in the form
^>,=^,-i + m,p^x, + Pl,; (3.3)
yk==y,-i^fn,p^'yu + P'iu> (3.4)
The positive directions of forces and displacements are indicated in
Figs. 2. 22, 2.23, and 2.24. The quantity 1^ represents the distance from the
point of application of the external exciting forces P^^ to the point of attach-
251
merit of the load to the elastic "beam.
From the condition of equilibrium of a section of the 'beam'(Fxg»2^2k) we
have
M'u-=-M^ + Y4^ cos o.^ - X^l^ sin a^.
(3-6)
To study the deformations, let us turn to FLg.2.3ff' which shows the k-th
section of an elastic beam Aj^B^ in a position of equilibrium and the same sec-
tion in a displaced position AjjB^* Let the quantities Xj,, y^^ ^k+i ^^^ 7k+i ^®
the displacement of the point Aj^ and Bj^ (ends of the section), and let ^^ ^.nd
^k+i ^® "^^^ angles of rotation of a tangent to the elastic axis on the left and
right ends. Furthermore, let h^ be the deflection of the beam at the k-th sec-
tion, i.e., the displacement of the right end of the beam (point Bj^) in a di-
rection perpendicular Aj^Bj^ relative to the tangent to the elastic aods at the
left end (point Aj^). Then we can write
•^A+i — -^A — ^k -r Va) sin a^\
^ji+i = ^fe + (8A + Vfe)cosa^,
(3-7)
where Ij^ and qli^ are, respectively, the length and angle of inclination of the
k-th section of the beam (Fig. 2. 2^).
Fig. 2. 22 Polygon of Forces Acting on a Section
of the Elastic Fuselage Model.
252
'^kP^Yk
Plg*2.2S Polygon of Forces
Applied to k-th Element of
an Elastic Fuselage Model.
^K±L^
1221
Fig. 2. 2^ Polygon of Forces Applied
to a Section of an Elastic Fuselage
Model.
Applying the usual methods of strength of materials, we find the following
equations correlating the forces and deformations:
-A
\m,-v\mI
K^l-^u+^h\
A&.= -
2Eh
■W,^A4]^].
(3.S)
(3-9)
(3-10)
The displacements of the team points Xj^ and y^ are related with the center-
of-gravity displacements of the loads by the evident formulas
ft^'ft-
(3.11)
The recurrence formulas {3*3)9 (3 '4), {3*5)9 and (3*7), together with
e<^.(3*S), {3*9)9 (3.10), and (3»ll) and if the forces and displacements of the
k-th load are known, make it possitile to determine the forces and displacements
of the (k+l)-th load. Using these formulas, we can solve the problem "by the
"chain method", as follows: After assigning the amplitudes s^y© and ^q at the
left end of the beam, it becomes possible to determine, successively passing
from section to section, the anplitiades and forces at the extreme right end of
the beam, expressing them in terms of the quantities Xq, Jq, ^q^ If the beam
has n loads, we can thus determine the quantities X^^, Y^, and M^ at the right
253
end or the "residual". However, since the right end of the iDeam is free, the
"residual" should "be equal to zero, i.e., at the right end of the "beam the con-
ditions
Xn^Yn^Mn=^Q
should be satisfied.
These conditions represent a system of three equations for determining the
unknowns Xq, yo» ^o» ^^ terms of which we had already expressed the vibration
amplitudes and the forces on all loads of the "beam.
This method of calculating the forced vibrations of a system (ELg.2.2l) is
conpletely analogoios to the well-known method of "residues" (Tolle method), /234
used for calculating torsional vibrations of multidisk systems (Ref*20). A
similar method is used for calculating flexural vibrations of elastic beams. In
the American and English literature such a method is known as Myklestad's method
(Ref.33, 34)- This method permits: l) finding the curve of dynamic stiffness
(Flg.2^11) of a system at any point and in any direction Idj calculating vibra-
tions at different values of p; 2) finding the natural vibration frequencies and
modes of a system from an analysis of the forced vibrations of the system close
to resonance, when the forced vibration anplitudes increase without bounds.
This method is especially convenient when using electronic conputers, with^
out which it is presently inpossible to conduct dynamic calculation in the
necessary volume.
For a practical application of this method it is convenient to express the
forces and displacements at the k-th section in terms of the values of Xq, yo,
and to give '^q in the form
where A^, Bj[, etc. are coefficients.
When calculating by the "chain" method, the values of these coefficients
of the k-th section must be used for determining their values for the (k+l)-th
section. Using recurrence formulas for forces and displacements, it is easy to
construct recurrence formulas for the corresponding coefficients. The follow-
ing formulas are obtained in this manner:
For the coefficients Aj^, By., G^, and T)^, we have
254
(3-12)
T
■^l-^U-
/*-i
fi/i
A— I
^^r
^Lr
4-1 sin o»-i
2£/*-i
(3.13)
Q Q Q
For the quantities Bi^, G^, and Di^, analogous formulas are obtained Idj replacing
the quantities A 1^ B, C, and D, respectively. This pertains also to the fol-
lowing formulas [eqs.(3*14) and (3-15)] •
For the coefficients ^4^, 5j, Cj and D^
k + l ft OPT. *
2EI.
lusin'iaf,
t\ sin a/.cosa;i
^r-i-
A^-lf,sinaf,Al.
For the coefficients A^^, B^^, C^ and Z)g:
A^ =A^A-
^l COS ak
2EI,
l\ cos a^sjsin a;;,
AM^
/ftCos2 a,,
A^ + lf,cosa^Al.
Ay —
(3.n4)
For the coefficients A^^ B^ ' C^ and D^l
D^ = D^-, + m.p^Dl - m,p^h,Dl.
For the coefficients A^, B^, C^andD^l
A^. = Ay-,+m,p'Ay+Pi^;
^l = By_, + m,p^B^,; ]
Finally, for the coefficients A^, B^, CfandD^l
j^M = A'^_^ + ^Li'*-i COS «;,_, - ^f_i/ft_, sin a;,_i + m^^p'Al -
- P" ipi^hl + /,) A\ - MO + PO^e,:
5^ = 5f_j + Bl_,l^_, cos «,_, - 5^_,/,_, sin a,_, +
+ m^h^p-^Bl - p-^ {m,fii + /,) B».
The formulas for C" and D" are obtained from the last equation on
ing the quantities B ty the quantities C and D, respectively.
(3.15)
mi
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
replac-
255
The derived formulas permit determining the values of the coefficients at
the next section from their known values in the previous section. Thus, moving
from section to section or from station
to station from left to right, the
values of the coefficients at the right
end of the beam are determined. On the
right (free) end (k = n), the condi-
tions
ymm
OM
^X-^
fx
^ 1
\1
5 xm
/
/
~— -
/
/
~0M
-OM
-OM
-0.03
-O.W
-OJl
Fig* 2. 25 Forced Vil^ration Mode of
an Elastic Fuselage of a Single-
Rotor Helicopter Ot)tained by the
Method of Residues.
(3.22)
should "be satisfied.
Solving this system, we find the
values of Xq, Jq, and «^o o^ interest
here:
Xo=-
Vq^-
K='
(3-23)
where
A = determinant of the system (3*22);
Axo, Ayo, A^ = determinants obtained from the determinant A by replacing
the corresponding colTamn by the free terms of the equations.
Knowing the quantities Xq, Jot ^^^ ^o pennies finding, by means of
eqs.(3.l2), the displacements and forces acting in each cross section of the
beam.
FigiH-e 2*25 shows the mode of forced vibrations of a single- rotor heli- /236
copter, determined by the indicated method. The vibration mode in this case
should be represented by three graphs: Xfc(x), yk(x), and «^ic(^)*
Table 2.1 gives the initial data for the performed calculation.
The forced vibrations were calculated from the following forces applied to
the rotor hub (load No.3):
where G is the helicopter weight.
256
No. of
Station
Xkim)
^k ( fe^/m-sec*)
0.10
-0.15
0,00155
1
1.7
1471
37.65
O.O0G6
1692
63.0
0.0068
4
7.2
3599
286.7
0.316
Table 2
.1
3
5
9.1
259
4.62
-0,15
6
12.4
24
7
8
9
10
5.1
13.7
15.1
16.7
18.5^
3584
24
22
92
153
260.5
0,19
O.IO
0.05
0.077
0,18&
0,778
0.207
0.0015
21
0.00042
0.00028
0.00017
0.00013
1.0
-
0.
43
-
One of the virtues of this method of calculation is that, for calculating
forced vi^brations it is not necessary to perform a preliminary calculation of
the natural vibration frequencies and modes of the system. Furthermore, in such
a calculation for different values of frequencies p, it is possible to construct
a graph of the dynamic stiff ness of the system D(p) at any point and also to /237
find all natural vi^bration frequen-
'600
'^00
'200
200
UOO
600
Fig,2.26 Curve of dynamic Fuselage
Stiffness, OlDtained by the Method of
Residues •
1
i 1
1 1
1 /
1
.
J"^ ^
/
: 1
1
1
I'tn
1
r\ '
1 I X
200
1
m 600 800 mo mo j
mo mo woo\20Do^pciicfi^
i
1 ' '
,
\\
cies and modes. Figure 2.26 pre-
sents the results of the calculation
of the graph of dynajidc stiffness
for the same system, given in
Table 2.1 for forced P^y • The
values of -p^ for which D(p) = give
the natural vibration frequencies of
the system, while the mode of forced
vibrations at a value of p close to
any of the natural vibration fre-
quencies Pk (k = 1, 2, 3, •••)
gives, with any desired degree of
acctoracy, the natural vibration mode
of this overtone. The modes of the
first three harmonics for the ex-
amined system obtained in this man-
ner are shown in Fig. 2. 27.
We note in conclusion that the method of "residues" presented here requires
performing the calculation with a very high acctiracy (at least four or five sig-
nificant digits). This makes the above method unsuitable in practice for a /238
keyboard calculator. However, as already indicated, vibration calculations in
the required voliome can generally be performed only on high-speed conputers for
which the indicated accuracy is conmon.
257
X]y
Pi^O P2^0
-1
;
First fundamental
1
1
1 A
. ia^
.a».5i?^
-- ^ec^r
jdj^<
—
—
^^
.^-^
r^ 1
Center of gravity
xm
1
Pr-
.p;^Z56 cyc/min
Xiy
10
Ps^pj-^mOcyc/mitt
First elastic overtone
/
>i-
Lyl
1
^A^
,^Xe
l--^pzp:=
i^-f
xm
^.y Pu = Pi = 1370 cgcf min
\ /
Second elastic overtony
yA
I
1
1
/
/"
2>^ 1 i.
/-
7-
""N
\
. ^~~r 1
15
1
\xm
1
Third elastic overtone
\
1
\
\
/
V
■y>
\
/
\
\
\
r'"
N
\ ^- /
/
L
■\y
10
r'f\
xm
!
\j
i
\J
— -
-5
a=--t-
Fig 2.27 Natural Vibration Modes of the Three Lower
Overtones of a Single-Rotor Helicopter Fuselage,
Obtained by the Method of Residues.
3. Consideration of the Effect of Shearing Deformation
All above-described methods of vibration analysis for a fuselage were based
on the use of conventional relationships of the strength of materials for bend-
ing of a thin beam. These relations take into account only tensile and coirpres-
sive deformation of the fibers of the beam material and disregard shear deforma-
tion. Furthermore, a consideration of these strains introduces certain correc-
tions into the calculation results, which are rather insignificant for the first
harmonic of vibrations (decrease in frequency by 5 - 6^), somewhat greater for
the second harmonic (decrease in frequency by 10 - 15^), still greater for the
third harmonic (20 - 30^), and so on. Therefore, if a vibration analysis re-
quires consideration of high harmonics, the vibration should be calculated with
consideration of shear strains caused by tangential stresses in the fuselage
skin. This can be performed in the following manner: If the calculation is
carried out for a model of the type shown in RLg^2.2l, then all formulas of the
"residue" method can be used, with the exception of eq.(3.8) which, in this case,
must be written in the form
Eh \
-^*+4-^*R8
;)■
{3.2k)
where 6^^ is the additional deflection of the k-th station due to the shear
force Qc
258
QA = -^ftSina^~rACOsajfe. (3-25)
This additional deflection 6 if can be determined by means of the following
formula [see, for exajiple (Ref.2l)]:
^*=5^''- (3.26)
Here, Y^ is the cross-sectional area of the fuselage at the k-th station, while
n is some dijnensionless coefficient determined "bry the formula
where
Ijc = moment of inertia of the cross section relative to the neutral
axis;
Sic(z) = static moment relative to the neutral axis of a part of the cross
section located above a straight line parallel to the neutral
axis and at a distance z from it;
6(z) = thickness of the fuselage skin at a distance z from the neutral
axis*
The integral in eq.(3*27) is taken over the entire cross section F of the
fuselage.
In conformity with the correction in eq.(3-^), corrections must be intro- /239
X Y ^
duced into the recurrence formulas for the coefficients Aj^, Aj^, Aj^, etc.
Section k* Combined Vibrations of the Sy stem Fuselage-Rotor
1. Vibrations of t h e System Fuselage- Rot or
The methods of calculating vibrations of elastic blades presented in Chap-
ter I assume that the blade is hinged to the hub which, in turn, is attached to
a stationary support. Actually, the hub is attached to an elastic fuselage and
forces are created during blade vibration that cause the hub to move so that, in
reality, the deflection at the hinge of the hub during blade vibrations is not
equal to zero but to the corresponding deflection of the fuselage.
Results of flight tests have shown in many cases that calculations of the
nattu?al vibration frequencies of blades performed without consideration of the
elasticity of the fuselage may result in substantial errors • In this connection,
M.L»Mil» has formulated and stated the problem of calculating combined vibra-
tions of the system fuselage-rotor as a single oscillatory system. The basic
results of investigations carried out in this direction are given below.
259
The frequencies and modes of natural combined vibrations of the system
fuselage-rotor can "be found by using the method of dynamic stiffness, whose es-
sence is presented in Subsections 2, 3f and 4 of Section 2.
However, performance of such calculations involves a large volume of conpu-
tational work# This pertains specifically to determinations of the lateral natu-
ral vibration frequencies of the system fuselage-rotor, when the dynamic stiff-
ness of the rotor in the plane of rotation is to be determined. Fiorthermore,
calculations show that the relation between fuselage and blade vibrations gener-
ally is weak and that the natural vibration frequencies of the system fuselage-
rotor can always be divided into two groups such that the frequencies of the
first groijp are quite close to the natural frequencies of the isolated fuselage,
in whose calculation the blade mass is considered as concentrated at the rotor
center, whereas the frequencies of the second groip are sufficiently close to
the natural blade frequencies calculated on the assimption that the blades are
attached to a perfectly rigid and infinitely heavy fuselage.
When the hub attachment to the fuselage is insufficiently rigid (elastic
rotor shaft, elastic reduction-gear frame, gear case), it may happen that some
of the frequencies of vibrations of the second group noticeably change in com-
parison with blade frequencies calculated by the usual method.
Therefore, the natural vibration frequencies of the first groi:^ can usually
be determined by means of methods presented in this Chapter as fuselage frequen-
cies, disregarding elasticity of the blades. An- exception are special cases
where, for exairple, the rotors are attached to light and elastic wings on a heli-
copter of side-by-side config-uration. In such cases, the frequencies of com-
bined oscillations of the system fuselage-rotor must be calculated with the
above-described method of dynamic stiffness.
As regards the natural blade vibration frequencies, it is apparently always
necessary to estimate the possible variation of some of these frequencies due to
local elasticity of the rotor attachment to the fuselage.
Thus, to allow for the correlation of fuselage and blade vibrations, it /2U0
suffices in practice to estimate only the change in natural blade frequencies
caused by local elasticity of the rotor attachment.
In the next Subsection, we will present a method for such a calculation to
determine the natural blade vibrations in the plane of rotation, with considera-
tion of the flexural elasticity of the rotor shaft. This case is the most im-
portant in practice.
To the elasticity of the rotor shaft one can always add the elasticity of
other elements of the rotor attachment (gear frame, gear case, etc.). Here we
will give certain iirportant fundamental considerations, from which it will be-
come obvious that only some of the natural -blade vibration frequencies are able
to change as a result of elasticity of the rotor attachment.
In Section 1 of this Chapter it was shown that not all harmonic conponents
of forces generated by vibrating blades "pass" to the fuselage, since many are
neutralized at the rotor hub casing.
260
For instance, diiring l:)lade vibrations of a five-blade rotor in the flapping
plane, the first foior harmonic conponents of forces transferred to the hub from
the blades (o), 2oi, 3(^, 4^) 3-1*^ neutralized at the hub and only the fifth har-
monic corponent is transmitted to the fuselage.
Hence it is obvious that, in calculating forced blade vibrations due to
forces corresponding to the harmonics co, 2^, 3oo, and 4tw,we must examine the
natural blade vibration modes and frequencies (with the method of e^ipansion in
natural modes), calculated for ordinary boundary conditions when the blade is
assumed to be hinged to a stationary hub.
When dealing with forced vibrations of the fifth harmonic, the presence of
combined vibrations of blade and fuselage must be taken into consideration.
The pl^sical meaning of this phenomenon is that the natural vibration modes
of a rotor with elastic blades can be divided into two groups:
1) rotor vibration modes at which the forces from individual blades are
neutralized at the hub casing;
2) rotor vibration modes at which the forces from individual blades are
summed at the hub casing and are transmitted to the fuselage.
Figure 2.2^, as a typical exanple, shows two such vibration modes for a
rotor with four blades since the picture is clearest for such a rotor. Both vi-
bration modes A and B correspond to the vibration frequency p x of a single-mode
overtone of an isolated blade in the flapping plane and differ only by the phase
distribution of the vibrations with
respect to individual blades. The
vibration mode A corresponds to a
situation where pairs of opposite
blades vibrate in opposite phase.
In this case, the forces pi, pg, pa
and P4 acting on the rotor hub mu-
tually cancel out at each instant
of time and are not transmitted to
the fuselage. The vibration mode B
corresponds to the situation where
all foixr blades vibrate in phase.
In this case the forces pi, Ps, Pa
and P4 are summed at the hub and
generate a force acting on the fuse-
lage and varying in time with a
frequency p^.
If the rotor hub is attached
to a perfectly rigid sipport, then
the frequencies of both vibration
modes A and B of the rotor are idenr-
tical and equal to the frequency pi
or to the natural vibrations of the first harmonic of an isolated blade with a
hinged butt. If the hub is attached to some elastic base with a vertical rigid-
ity c, the frequency of the vibration mode A will not change and remains equal /241
to pi, whereas the frequency of the modes B will decrease and that the more the
lower the rigidity c.
261
Fig. 2. 28 Vibration Modes of a Rotor
with Elastic Blades.
It can te demonstrated that the modes of the two indicated types exist for
a rotor with any n^umlDer of "blades z» These vibration modes can be characterized
by a formula • For exanple, all vibration modes of the z-bladed rotor corre-
sponding to the k-th overtone of vibrations of an isolated blade are character-
ized by the following law of blade vibration:
yn{x, t)=y},{x) cos s ifn cos Pftt (4*1)
where
y^(x, t) = deviation of a point with the coordinate x, belonging to the
n-th blade;
cos s^jf^ = characteristic of the law of vibration phase distribution
for individual blades, i.e., of the vibration mode of the
rotor as a whole;
s = any integer that can be called the order of a given rotor
vibration mode (s = 1, 2, 3, •••, z).
The quantities ^^ ^^^ determined by the formula
z
On the basis of eq.(1.13) in Section 1, it is easy to show that the vibra-
tion modes of the orders s = 1, 2, 3* •• -^ z - 1 correspond to a situation in
which the forces generated by individual blades are equalized at the hub and
that only the mode of the order s = z corresponds to a situation where the forces
from individual blades are summed and transmitted to the fuselage.
The modes A and B presented in Pig. 2. 28 are modes of the second and fourth
order for a four-blade rotor. It is obvious from the aforesaid that the natu-
ral vibration frequencies of a rotor, corresponding to vibration modes of all
orders with the exception of s = z, do not depend r^Don the elasticity of the
hub attachment and that only the frequencies corresponding to the rotor vibra-
tion mode of the order s = z depend on this elasticity.
We can further show that all harmonics of forces that excite blade vibra-
tions in the flapping plane, with the exception of the "transient" harmonics zuo,
2za), 3za), etc. will excite only those rotor vibration modes at which the forces
produced by the blades are neutralized at the hub and that only the harmonic 72^2
conponents of the exciting forces corresponding to the "transient" harmonics
wili excite rotor vibration modes at which the blade-generated forces are simmed
and transmitted to the hub.
Hence, we can draw a useful practical conclusion: If we construct an ordi-
nary resonance diagram of the blade (see Fig. 1.6 of Ghapt.I) in the flapping
plane, calculated without consideration of elasticity of the rotor attachment to
the fuselage, then the resonances with all of the harmonics, except for reso-
nances with the harmonics zuo, 2za), etc., correspond to reality. The resonances
with the harmonics zuo, 2zuo, etc., must be investigated additionally, taking in-
to account the elasticity of the rotor hub attachment and refining the values
of the corresponding natural frequencies.
262
However, it should be mentioned that, in studies of blade vibrations in
the flapping plane, it is generally possible to disregard the elasticity of hub
attachment for such harmonics since the rigidity of the hub attachment in a
vertical direction is usually large and has little influence on the natural blade
vibration frequencies (except for the case of rotor attachment of a side- by-side
helicopter to light and flexible wings) •
In studying the resonance diagram of a blade in the plane of rotation the
effect of elasticity of the rotor hub attachment to the fuselage must be taken
into consideration. All above considerations hold also for blade vibrations in
the plane of rotation, with the only difference that in this case the "tran-
sient" harmonics are the harmonics (z - 1)ud, (z + 1)cd, (2z - 1) o), (2z + l)cjo,
etc. Furthermore, at resonance with the harmonics zoo, 2za), etc. in the plane of
rotationj allowance must be made for the combination of rotor vibrations with
torsional vibrations of the transmission system (the pertaining calculations can
also be carried out on the basis of the method of dynainic stiffness) .
2* Calculation of the Natural Rotor Blade Vibrations in the Plane
of Rotation , w ith Consideration of Elasticity of the Rotor
Shaft and Attachment to the Fioselage
Let us examine the problem of natural blade vibrations of a rotor mounted
to a flexurally elastic shaft (Fig. 2. 29). Let the rigidity of the shaft with
respect to the force P applied to the shaft at the hub center and lying in the
plane of rotation of the rotor be equal to Cq* Consequently, the force P and
the resultant displacement 6 of the shaft end are correlated by
P-Co6, (4.2)
In this case, it is immaterial whether the displacement 6 is produced by bend-
ing of the shaft itself or is due to the elasticity of its attachment to the
fuselage.
Let us discuss only the case when the given rigidity is identical in all
directions in the plane xDz, i.e., when the elastic sxjpport to which the rotor
is attached is isotropic. In reality this is not so, but the rigidities of at-
tachment in the directions of the Ox and Oz axes generally differ little so that
the sipport can be assiomed as isotropic, understanding by the quantity Cq the
arithmetic mean of the rigidities Cx and c^:
.Cx±Cz
(4.3)
Calculation of natural vibrations of a rotor on an elastic base can be per-
formed by the method of dynamic stiffness.
First, we introduce the concept of dynamic stiff ness of a blade in the plane
of rotation. Let a flexurally elastic blade in the central centrifugal force
field be attached at the root by a hinge such that the hinge is able to move /2U3
freely in a direction perpendicular to the axis of the undeformed blade (see
Fig. 2. 30).
263
[JUL
U=-UgCQSpt
Fig. 2. 29 Diagram of Rotor on
Elastic Shaft.
Fig.2«30 Diagram for Calculation
of Forced Blade Vibrations to De-
termine Dynamic Rotor Stiffness.
Furthermore, let the blade execute steady forced vibrations under the ef-
fect of a lateral exciting harmonic force
F ^Fq cos pt,
applied to the hinge A. In this case, the point A of the application of force
will also execute vibrations according to the law
u = Uo cos pt.
We will call the quantity
Uq
(4-4)
the dynamic stiff ness of the blade.
The dynamic blade stiffness can be determined either by the method given in
Subsection 2 of Section 2 or by the method of auxiliary mass (Sect. 2, Subsect.4)»
In so doing we must take into account that the blade moves in a centrifugal
force field so that it is no longer a question of solving an equation of the
type of eq#(2»2), as had been done in calculating the fuselage, but of solving
the equation of blade vibration in the plane of rotation (see Chapt.I, Sect.l,
Subsect.ll), which has the form
Here, N is the centrif-ugal force in the blade section at a radius r-
(4.5)
When using the method of auxiliary mass, the natural blade vibration fre-
quencies and modes miost be calculated in the plane of rotation in the presence
of an attachment according to the scheme depicted in Fig. 2.30, with a different
value for the auxiliary mass Ami^ at the point A, using the method presented in
Chapter I, Section 2, Subsection 5.
264
From the results of such a calculation we can construct the graph of Z^^ =
= f(p). An exanple of such a graph is shown in Fig. 2.32. The points of irv-
finite discontinuities of the function f(p) give the natural frequencies of the
blade with a fixed hinge at the
point A, i.e., the natural fre-
quencies of a blade for the
case of an infinitely large
rigidity of the rotor shaft.
The points at which Am^ =
yield the natural frequencies of
a blade attached freely accord-
ing to the scheme depicted in
Fig. 2.30.
The magm-tude of the dy- /2hU
namic blade stiffness correspond-
ing to this value of p can be
deterndned from the formula
Z),(/?)^-(/;2^c.2)A;7Z,(p). (4.5a)
The additional term uu^Am^Cp)
in this formula, is due to the
centrif^agal force coirponent of
mass Am^ directed along the
Fig. 2. 31 For Calculation of Dynajnic Stiff-
ness of a Rotor with Elastic Blades.
normal to the blade.
We will show further that the dynamic stiffness of the rotor as a whole can
be found if the dynamic stiff ness of the blade is known. Let us turn to Fig. 2. 31
which gives the planform of a rotor hub with vertical hinges and the k-th elas-
tic blade. Let xOy be a coordinate system rotating together with the rotor with
an angular velocity uo. Furthermore, let the center of the hub execute pre-
scribed harmonic vibrations in the plane of rotation in obedience to the law
':]
(4.6)
Such vibrations of the hub cause vibrations of the elastic blades in the
plane of rotation, reducing the problem to finding the forces exerted by the vi-
brating blades on the hub diiring its motion.
Let us choose an auxil±ary rectangular coordinate system rotating together
with the rotor nOr, for which the Or-axis is parallel to a straight line passing
through the center of the hub and through the drag hinge A of the k-th blade.
The Or-axis maices a certain angle tk with the Ox-axis. We denote by Uq and Vq
the coordinates of the hub center in the system nOr. Then, obviously.
265
During vibrations of the hub in accordance with the law (4»6), the coordi-
nates Uq and Vq will vary in time in obedience to the law
«o = ( -^0 sin %) cos pi + (y^ cos ^g sin pi;]
'^o = (-^oCOs6jcos/?^ + (^oSin^A)sin/;if. J {k-7)
Furthermore, let u denote the deflection of the point of the elastic /2U5
blade axis at a radius r from a straight line passing through the drag hinge A
of the blade and running parallel to the Or-axis. During vibrations of the
blade, the quantity u is a function of the radius r and the time t such that u =
= u(r, t).
Let w be the vector of the total acceleration of a point of radius r of the
elastic blade eods* Then,
where
^rei ~ vector of relative acceleration of a point due to motion in a
moving coordinate system nOr;
Wtr = vector of translational acceleration due to motion of a point to-
gether with the coordinate system nDr;
Wcor = vector -of Coriolis acceleration.
We then introduce the lonit vectors i and j, directed along the axes r and
n, respectively. Then, we can write
If we denote by w,^ and Wj. the projections of the vector of total accelera-
tion onto the axes On and Or, the following expressions are obtained:
The equation of equilibrium in the centrifugal force field has the form
iE/ur-{Nuy=^q, (4.9)
where q is the intensity of the lateral load applied to a beam.
Dioring blade vibrations, the lateral load due to inertia forces can be
written in the form
q (r, t)=--~ mw^ =^~m [{Uq — ^i^a^) + (ii — to2^) + 2(ox;o] ,
266
where m is the linear mass of the blade [m = m(r)].
SulDstituting this expression into eq.(4»9), we ©"btain the following partial
differential equation for determning the function u(r, t) :
(^/O"-(A^«T + wi£-a)2a = ^*(r,0, (4.10)
where
^*(r,0=-/;i[^o-^'>'^o + 2a)io]. (4-11)
If the motion of the center of the hut is given hy ecp.(4«7), then the load
q^^(r, t) will t)e a known function of time.
The unknown function u(r, t) should satisfy eq.(4*lO) as well as the
"boundary conditions
u (0,^)=w" (0,0=0; 1
u"{R, t) - {EIay\r^R = 0. J ih^^^Z)
Differentiating eqs.(4*7) and substituting them into eq.(4-ll) will yield
q^{r, i)^mAf^cos pt + mBf^sinpt, (4*13)
where the constants A^ and Bj^ are determined by the formulas
The solution of eq.(4»lO), corresponding to steady forced vibrations due /246
to a load [see eq.(4«13)], is sought in the form
u{r, t)=li{r) [Af^cos pl-i-B f^sin pi]. (4.15)
Substituting this e:xpression into eq.(4*lO) with the right-hand side for q"^'"
fixDm eq.(4«13)* we find that the function u(r) should satisfy the ordinary dif-
ferential equation
(£/«")" - (Nuy - (;,2+a,2) mu=m, (^,16)
as vrell as the t)Oiindary conditions
«(0) = «"(0)=0;
(4.17)
«"(/?) = (£/a")'|;.«=0. '
We note further that, in calculating blade vilDrations excited t)y vibration
of the hinge A according to the scheme depicted in Fig. 2*30, it is necessary to
solve an equation of the form
267
where u is the total displacement of a point of the elastic "blade axis of radi-
us r. In this case, the function u(r, t) should satisfy the conditions
ri(0, t) =^Uq cos pt;
a"(/?,/)^0;
iE/uy\r^j, =0.
Seeking the solution of this equation in the form
u =■ [Uq -^li (r)] cos pt,
we arrive at the conclusion that the fiinction u(r) should satisfy the equation
which differs from eq.(4«l6)_only tj the constant Uo(p^ + cjo^). The boundary
conditions for the function u(r) in this case fully coincides with eqs.(4*l7)«
Thus, during blade vibrations according to the scheme shown in Fig #2.30,
the function u(r) is the same as in the problem of interest here [see eqs.(4-l6)
and (4«17)] ii* we select the amplitude Uq such that the condition
Wo(p2-Kco2) = l (4.18)
is satisfied.
Physically, this means that the mode of forced blade vibrations in the
problem of interest here coincides with the mode of blade vibrations excited ac-
cording to the scheme in Fig. 2. 30. On the basis of this result, an ijiportant
formula is derived. For this, we note that during vibrations of a blade attached
according to the scheme shown in I^g.2.30 and excited by the force F = Fq cos pt,
the sijm of the projections of all lateral inertia forces applied to the blade /247
should be balanced by the force F. Hence, we find
f =i,mi,{p){p^-{-<^)Uo=~-{p^+'>>'^) (■ m{Ua-\-ti)dr=
in
==„(^2^o)2)^ioWi-(/?2+a)2) J madr.
y.h
where m^ is the blade mass -up to the drag hinge
268
niu
r mdrV
On satisfying condition (4.18), we olDtain the formula
mudr^- ^^t-^^t
/?2 4- 0)2 *
(4.19)
meaning that the integral with respect to the "blade of the fimction mu [where
u is the solution of eq,(4.l6)] is e:xpressed in terms of dynamic blade stiffness
or, which comes to the same, in terms of the auxiliary mass Ami^(p).
It is now easy to obtain expressions for the forces exerted on the hub by
the vibrating blades. We denote by Qi^ and N^, respectively, the projections on-
to the On- and Or-axes of a force exerted by the k-th blade k on the drag hinge
of the hub. Then,
r mwr dr=^ — \ m{u — ni^u) dr ■
'y.h
R
R R
-^0)2 ^ mrdr-\-2iii C ma dr.
(4-20)
Substituting here eqs.(4»15), (4*7)> (4»14) and taking into account
eq.(4*l9), we find
Qu = A^i { [(/?2 4- 0)2) Xq + 2to;?ro] sin % cos pt —
— [(/?2 + 0)2) y^ -I- 2^pXq] cos 'I'^ Sin pt];
Nu = N,-
(p2 4- 0)2)
COS ^l; j^ COS pt-\-
slnC^^sinpi^
(4.21)
(4.22)
where Nq = uo^ j mr dr is the centrifugal force exerted by the blade on the drag
hinge.
Denoting by X and Y the forces exerted on the hub by the vibrating blades,
269
we derive the formulas:
/2^8
^= y^i-Qk^in^k+^k^os^^);
ft-i
Substituting here eqs.(4»2l) and (4«22) and taking into account the proper-
ties of the trigonometric sums descriTDed in SulDsection 2, Section 1 of this
Chapter [eqs.(l.l3)]* we arrive at the following e:xpressions:
X
-{■
y AOTi [(/?2 + 0)2) xo + 2o^pya] + -~-
-A/n,2;7(Oi,o-^^^-4;;Vxo'
/?2 J- 0)2
cos /?/;
r=
•A^,[(/?2 + aj2)^^ + 2co/?Xo] +
+f
/72 -^ 0)2
sin /7^.
(4.23)
(4.24)
On the other hand, we can construct the equations of motion of the hub of
a rotor on an elastic shaft, which have the form
where
mjju^ = mass of the hub casing;
Cq = shaft rigidity ♦
If the motion of the hub takes place in obedience to the law (4*6), the
last equations will yield
-^ = (/?^U - (/?^ + ^^) ->^o " 2oj;7£/o] + CqXo} cos /?/;
^ = [f^hubl — {P'^ + ^^) yo — 2o3ji?Xo] + Coi/o } sin pt.
If we equate these eij^ressions to eqs.(4»23) and (4»2^), we obtain a system
of two Hnear homogeneous equations for determining the airplitudes Xq and JqI
(4.25)
where
270
A = (p^-i-<,^)\^{m,-~^m,) + m^
_2^Am,H:^
{p2 4- 0)2)
(4.26)
(4.27)
Equating to zero the determinant of this system, we obtain the character-
istic equation for determining the natural frequencies p:
A B
B A
:^2_^2^0,
(4.2S)
whence
^ = -4-^.
In the case A = -B [as is apparent from eq.(4.25)]^ Xq -Jo* This cor- /249
responds to rotation of the hulD center in the direction of rotation of the rotor
[see eq«(4.6)] .
gC^m^kS _
ZOOOpcj/e/m in
Fig. 2. 32 Determination of ViToration Frequencies of
a Rotor on an Elastic Shaft, by the DyToamic
Stiffness Method.
In the case A = B we have Xq - -Jo, which corresponds to rotation of the
rotor center opposite to the rotation of the rotor.
The characteristic equations (4.2S) can be solved with respect to the quan^
tity Am^(p). This yields the following equation:
Lm,{p)= - ^2 7
2z/?2to2
/?2 -j. 0)2
Z 2^D2a,2
(4.29)
271
This equation can "be solved graphically by superinposing, on the curve of
the auxiliary blade mass Am^ = Ain^(pj, two curves corresponding to the right-
hand side of this e^^ression in which we take either the upper signs (minus sign
in the numerator and plus sign in the denominator) or the lower signs. The
first of these quantities will be denoted by Ami(p) and the second, by AmgCp).
The abscissas of the intersection points of the curve Am^Cp) with the graph
of auxiliary blade mass Am^j(p) yield the natural vibration frequencies of a
rotor on an elastic shaft, corresponding to vibration modes in which the center
of the hub rotates in the direction of rotation of the rotor, with an angular
velocity p relative to the coordinate system xOy fixed to the rotor and hence
with an angular velocity p + co relative to the body-fixed coordinate system
(helicopter body). Obviously, such modes can be excited only by the harmonics
(z - l)ao, (2z - l)a), etc. The abscissas of the intersection points of the
curves Am2(p) and Am^(p) yield the natural vibration frequencies of a rotor on
an elastic shaft in which the hub center rotates in a direction opposite to that
of the rotor. Such vibration modes can be excited only by the harmonics (z +
+ l)uo, (2z + 1)00, etc.
Figure 2*32 gives the graphs for the curves Am^(p), Amx(p), and Am3(p),
constructed for the following initial data: Cq = 500 kg/mm; m^^ t = 3S kg*cm /m;
m^ = 15 kg»sec^/m; udq = 190 ipm; z = 5* These graphs show appreciable differ-
ences between the natural frequencies of a rotor on an elastic shaft and the
natural frequencies of an isolated blade. For exairple, the point H of an in- /250
finite discontinuity of the curve tjn-^(p) corresponds to the frequency of a
single-node overtone of natural vibrations of an isolated blade of the given
rotor (with a stationary hub). In this case, p = Pi = 640 cycles/min. Here,
Fig. 2.33 shows the vibration mode of this overtone.
a)
y.
p=Pi
'D,15
7^ ff^^Mi^^uH^^
.^i,
Fig. 2. 33 Modes of Blade Vibrations,
a - Mode of blade vibrations without consideration of
shaft elasticity; b - Modes of blade vibrations with
consideration of blade elasticity.
In addition to this natural frequency, a rotor on an elastic shaft also has
vibration frequencies corresponding to the points A, B, C and D of the inter-
cepts of the curves Ami(p) and AmgCp) with the curve k^-^{-^) • Here, vibrations
of modes corresponding to the points A and D can be excited only by the har-
monics (z - l)a), (2z " 1)00, etc- (in this case, 4tu and 9tioetc.). Vibrations cor-
responding to the points G and B can be excited only \>j the harmonics (z + l)u),
(2z + 1)00, etc. (in this case, 600, lluo, etc.).
272
These resonance cxjrves were plotted for a helicopter which first had "been
equipped with a four-lDlade rotor; however, later the rotor hub had to he modi-
fied and the rotor was designed as a five-hlade type so as to eliminate the se-
vere resonance of the blade with the harmonic 3^ in the plane of rotation
(point A) .
Figure 2^33 gives the natural blade vibration modes in the plane of rota-
tion, with consideration of shaft elasticity corresponding to the points A (p^ =
= 560 cycles/min) and B(p^' = 76I cycles/min).
In conclusion, we should mention that the above method for determining the
natural frequencies of a blade in the plane of rotation with consideration of
shaft elasticity is one of the most conplex exanples of using the method of dy-
namic stiffness; this was the main reason for describing it here in some detail.
As regards finding the natural blade vibrations in the flapping plane with con-
sideration of elasticity of the hub attachment and of the blade vibration fre-
quencies in the plane of rotation with consideration of torsional elasticity of
the transmission system (which are excited by the harmonics zcjo, 2zu>, 3zuo, etc*)*
the calculations involved are much sinpler and can be carried out in full on the
basis of the principles set forth in Section 2»
273
CHAPTER III /251
GEDUM) RESONANCE
Qround resonance usually is to mean spontaneous vibrations (build-up) of a
helicopter on the ground "with increasing anplitude* This phenomenon was first
noticed after a drag hinge permitting the blade to nove in the plane of rotation
of the rotor was introduced into the design of the rotor hub#
In the history of helicopter engineering there were quite a few cases where
a helicopter was destroyed by vibrations of this type. Attenpts to eliminate
ground resonance on a full-scale helicopter sometimes required extensive modifi-
cations of the helicopter design. This forced design engineers to work on the
development of the theoxy of ground resonance and reliable methods of its calcu-
lation, which would permit selecting the characteristics of the structural mem-
bers determining the stability margin of the helicopter on the ground.
At present there is a theory of ground resonance which explains all the
most iir5)ortant features of this phenomenon and permits calculating the design
characteristics on which depends ground resonance. This theory arose as a re-
sult of numerous theoretical and experimental investigations of ground resonance
carried out both in the Soviet Union and abroad. Of the Soviet works on the
theory of ground resonance we must point out first the works of B.Ya.Zherebtsov
and A.I.Pozhalostin.
Investigations of ground resonance have shown that the physical essence of
this phenomenon involves the following: During natural vibrations of the rotor
blades in the plane of rotation (relative to the drag hinges), which can arise
from any impetus (wind gust, rough landing, etc.), inertia forces appear in the
plane of rotation of the rotor. Being transmitted to the helicopter fuselage,
they cause its vibrations on the elastic landing gear. The forces swinging the
helicopter vary with a definite frequency depending upon the natural frequency
of the blade in the plane of rotation and the angular velocity of rotation of
the rotor. A helicopter is most easily swung when the frequency of change of the
exciting forces is close to the frequency of natural vibrations of the helicopter
on an elastic landing gear. Simultaneously with vibrations of the helicopter
body, forces arise which swing the helicopter in the plane of rotation. The
presence of this bilateral couple between vibrations of the helicopter and blades
results in the helicopter becoming unstable at a certain angular velocity of
rotor rotation, i.e., the helicopter vibrations once begun (as a consequence of
some iopetus) are not danped but increased.
The basic means of combatting ground resonance are:
1) The installation of special danpers on the drag hinges of the rotor
blades which damp the blade vibrations in the plane of rotation.
2) The introduction of special danping elements in the design of the /252
274
shock a'bsorber strut or the proper selection of the characteristics of hydrau-
lic resistance of the shock ahsorher struts in forward and reverse strokes, and
also the characteristics of rigidity of the shock absorber struts and pneumatic
tires •
The proper selection of the characteristics of the blade danpers and the
characteristics of the rigidity and danping of the landing gear is the main
purpose of calculating a helicopter for ground resonance •
The theory of ground resonance which will be presented below holds true only
for rotors with a number of blades n ^ 3»
The theory of ground resonance of a two-blade rotor has a number of special
features and is appreciably more conplex (Ref •36).
Section 1- Stability of Rotor on Elastic Base
1. Statement of Problem and Equations of Motion
The most iirportant featiores of ground resonance of a helicopter can be ob-
tained from an examination of the motion of some idealized mechanical system,
which we will call a "rotor on an elastic base" •
Such a system is schematically shown in Fig.3*l«
l^^f^ The shaft of the rotor with heavy and perfectly
— J rigid blades (3), attached to the rotor hub by
zzzzzzza means of the drag hinges (4)^ rotates in sup-
7- / ^ J jj - rr/ ' V
PP7> R^ L
.^v&l^ ports rigidly connected with some heavy casing
'~^ (body) (l) which is elastically mounted to a
^c^ stationary base (2) and has only one degree of
2 freedom, namely forward displacement along the
r^ axis Ox parallel to the plane of rotation of the
I rotor. Upon displacement of the body (l) along
the Ox-axis, an elastic restoring force is gen-
Fig .3. 1 Diagram of Rotor erated by the spring c and a danping force by
on Elastic Base. the danper k. Let us assrone the elastic and
1 - Casing; 2 - Base; 3 - danping characteristics of the base to be linear.
Blade; 4 - Hinge. i.e., that the force X acting on the casing (l)
during its displacement x(t) is expressed by the
formula
X=~CX~K — , (1.1)
dt
where
c = coefficient of stiffness of the spring (spring constant);
k = danping coefficient.
We will call the quantities c and k the coefficients of stiffness and dajip-
ing of the elastic base. If hiq is the mass of the casing (I) and P^ is the pro-
jection onto the Ox-axis of the force exerted on the casing by the rotor, then
the equation of motion of the casing can be written in the form
275
mQX~\-KX-{'CX^=^Pj^
(1.2)
Here and "below, the dots denote differentiation with respect to time.
Furthermore, we will assume that the rotor rotates uniformly with an angu-
lar velocity uo in vacuum, i.e., we will neglect the aerodynamic forces. The
theory of ground resonance disregarding aerodynamic forces agrees rather well
with experiment. Thus, only inertia forces arising during blade vibrations in
the plane of rotation are taken into account.
To construct the equations of motion of the "blade, let us turn to Figs. 3*1
and 3.2. /253
Let us select a stationary rectangular coordinate system Oxyz. The axis Oy
is directed along the axis of the rotor shaft, at a position of the casing (1)
corresponding to static equilibrium. The direc-
tion of the Ox-axis is taken such that the only
possible displacement of the casing is directed
along the Ox-axis.
1 i
y im >
k\
\/i.
<K
/^
A^kk
\ v<>^ ■
X^ X X'
As usual, let x be the displacement of the
axis of the rotor shaft together with the casing
along the Ox-axis (Fig .3. 2). Furthermore, let
■^y, be the azijuuthal angle of the k-th rotor blade
reckoning from the positive direction of the Ox-
axis.
The angles ^y^ of different rotor blades are
determined by means of the formula
Fig .3. 2 For Derivation of
Equations of Motion.
n
(1.3)
where n is the number of rotor blades; k = 1, 2, ..., n.
We denote by ly.h ^^® distance AB (Fig .3. 2) from the axis of rotation A to
the axis of the vertical or drag hinge B, and by Z^^ the angle of deflection of
the k-th blade during its rotation relative to the drag hinge, taking 5ic as posi-
tive when the blade is deflected in the direction of rotation of the rotor.
Then, the coordinates Xj^ and z^ of the element of the k-th blade with a
mass dm at a distance p from the axis of the drag hinge are expressed by the fol-
lowing formulas:
(1.4)
Differentiating these e:^ressions twice with respect to time, we obtain
formulas for determining the. conponents of the acceleration of the blade element;
276
^ft - - ^''U Sin ^j, - Q (CO + k^f sin (6^ + $,) + 5, Q cos (^^ -f S,).
In deriving the equations of small "blade vil^rations relative to the drag
hinge we must, as usual, limit ourselves to small quantities of the first order #
Therefore, we can assume that
Thus, with an accuracy to small quantities of the second order, the formu-
las for the accelerations Xj^ and y^ can be written in the form
X^=X^ 0.2 1^^ cos %~Q (0)2 + 2a)g COS {^^ + g + \q sin (0^ + y ; 1
When the system moves in a vacuum, the rotor blades at each instant of
time t are loaded only by inertia forces. The elementary inertia forces acting
on a blade element are expressed by the formulas:
In the drag hinges of the rotor hub, let there be linear elastic and damp- /254
ing devices which, during rotation of the blade relative to the drag hinge, load
it by the moment
directed toward the side opposite to the positive direction %^^ . We will call c^
and k^, respectively, the coefficients of elasticity and dairping of the blade.
At each instant of time, the moment from the inertia forces applied to the
blade relative to the drag hinge should be balanced by the moment M. Therefore,
we can write
j [^AQsin(6;,-fy-z,QCOs(^^ + g]^/7i-=c,^;, + K^^ft,
where integration is carried out over the blade length I .
The equation of motion of the k-th blade is derived from the last expres-
sion and from ecp.(1.5) after sinple transformations. Since we are interested
in the equations of small blade vibrations, we can l±mit ourselves to terms of
the first order of smallness relative to the quantities x, x, 5jc» ^"^ Ik* after
discarding terms containing squares and products of these quantities. Then, we
can put
cosSft;==:i"l;
277
sin {^^ + y ^ sin % + S^ cos 6;^;
cos (^ft + g ^ cos tj>;^ - $ft sin (f ;^.
After such sinplifications, the equation of small vibrations of the k-th
iDlade will take the following form:
'^k + 2n,i, + (pl + vy)k,=^~-^ i^sin^,. (1.8)
Here, the following notations are tised:
k
nv =
b
21..
= relative danping coefficient of the blade;
h
p2 - 5 — = natural frequency of a nonrotating blade (at uu = O) rela^
ly. h tive to the drag hinge;
Vq = dimensionless blade parameter determined by the formula:
(1.9)
where
S^^^ = fpdm = static *blade moment relative to the drag hinge;
('
^v.h ^ Jp ^dm = moment of inertia of the blade relative to the drag
^ hinge.
The right-hand side of eq.(1.8) represents the moment due to inertia forces
acting on the blade, generated by the rotor shaft displacement (x) . When the
shaft is stationary, at x = 0, eq.(1.8) describes the natxoral blade vibrations
of a uniformly rotating rotor in the plane of rotation.
The general solution of eq.(1.8) without the right-hand side has the form /255
S,=$,,^'^*'cos(/7,^^cp,),
where ^^^o ^^^ ^ic ^^® arbitrary constants, while the quantity p^ is determined by
the formula
and represents the angular frequency of natural blade vibrations in the plane of
rotation.
Furthermore, it is necessary to determine the force P^ exerted on the casing
278
by the rotor* The force P^ represents the resultant of the inertia forces of
vibrating blades and, on the basis of the well-known theorem of motion of the
center of inertia (center of gravity) of a mechanical system, can be determined
as the product of the mass of the blade system and the coiiponent of acceleration
of the common center of gravity of the blade system along the axis ox«
Let us derive formulas for determining the coordinates of the common center
of gravity of the blade system.
Let Xv and z. be the coordinates of the center of gravity of the k-th
c k Q
blade. Then, the coordinates Xc and Zq of the center of gravity of the blade
system can be calculated by means of the expressions:
n
1 V^
^k^
(1.10)
Furthermore, let p^ be the distance of the center of gravity of the blade
from the axis of the drag hinge. Then, in conformity with eqs.(1.4)^ the coor-
dinates Xc, Zc can be determined as
Xu^ -X + 4, COS6, + Q,COS(6, + g;
Substituting these expressions into eq.(l.lO) and considering that, for
n s: 3 [see Chapt.II, Sect.l, Subsect.2, eq.(1.13)].
y cost;>^=0;
A"i
2 sin 6^ = 0,
(1.11)
we obtain the following sinple expressions for the coordinates of the common
center of gravity of the blade system:
n
fe-1
(1.12)
279
The force P^ acting on the elastic "base can 'be determined from the formula
Twice differentiating the first equation of the system (1.12), we obtain /256
Substituting this expression into eq,(1.2) will finally yield the follow-
ing equation of motion of the casing:
n
This equation is conveniently written in the form
n
jc -L 2 «oi + pIx = -^ ^li^k- <»'y sin 6^ + 2o4 cos 0^] ,
where the quantity
M=^mc^ + nmi, (1.13)
represents the total mass of the system, while no is the relative dairping coef-
ficient of the elastic base, determined by the formula
2M
(1.1^)
and the quantity po represents the angular frequency of natural vibrations of a
rigid rotor (without drag hinges) on an elastic base and is determined by the
formula
^o^ir* (1.15)
We will now write the system of equations of motion of a rotor on an elastic
base, consisting of the equations of motion of the blades [eq.(1.8)] and the
equation of motion of the casing of the base:
i, + 2n,i + (pl + v2<o2) s^ = _^ X sin ^,;
x-\-2nok^plx = -^^[(i^-'^'i,)sin<^,+ 2<^i^cos-]>,].
M
*-l
where k = 1, 2, •••, n.
280
(1.16)
Thus, the equations of small vibrations of a rotor on an elastic "base repre-
sent a homogeneous system of (n + l) Unear differential equations with periodic
coefficients for determining (n + 1) unknown functions x(t), 5ic(t) (where k -
= 1, 2, ..., n).
2* Stability Analy s is and. Basic Results
Investigations conducted tiy Coleman (Ref .35) and B#Ya.Zherebtsov showed
that, for a rotor with a number of blades n s 3, this system of equations can be
reduced to a system of linear equations with constant coefficients, if we re-
place 5k (t) by new variables x^Ct) and ZcC'^^) representing the coordinates of the
center of gravity of the blade system. In the case of a two-blade rotor,
eqs.(l.l6) cannot be reduced to equations with constants* An investigation of
the stability of motion of a two-blade rotor on an elastic base is quite coirir-
plex. Its presentation can be found elsewhere (Ref .36). B.Ya.Zherebtsov studied
also the case of a two-blade rotor on an isotropic elastic support when the /257
casing of this sipport had two degrees of freedom - in direction of the Ox- and
Oz-axes (see Fig .3 -2) - and the stiffness of the base in both directions was
identical. In this exceptional case, the problem is easily reduced to a system
of equations with constants.
Here, we will investigate the stability of a rotor with a nijimber of blades
n ^ 3, which is of the greatest practical value.
In order to obtain the equations of motion with constant coefficients, we
will transform eqs.(l.l6) to the new variables x(t), Tl(t), Z{t) related with the
previous formulas;
n
(1.17)
The new quantities T] and Cj ^s is apparent from eqs»(l.l2), are equal - with
an accuracy to within the constant factor Pc/n - to the coordinates of the center
of gravity of the blade system in a moving coordinate system x'Az' whose axes
are parallel to the Ox- and Oz-axes of the fixed system, while the origin of the
coordinates A coincides with the center of the rotor (see Fig. 3*2) •
To derive the equations of motion in the new variables, all equations of
motion of the blades [the first equation of the system (1.16)] must first be
multiplied by cos ^^ followed by addition of their left- and right-hand sides
from k = 1 to k == n; multiplication is then performed by sin ^^ again followed
by addition. Here, it miist be noted that, for a rotor with a number of blades
n ^ 3, we have by virtue of eqs.(1.13) of Chapter II:
2 sint!>;^costJ>A^O;
1
(1.18)
281
Pu3rbhermore,
2sin^1-*=f .
ft-1
2 SftSin^ft^Ti— coC;
ft-l
/I ..
'^ .. ...
(1.19)
The last formulas are obtained 'bj successive differentiation of eqs.(1.17).
This results in the following system of equations:
(1.20)
2 I,./,
I + 2« 'C - [co^l - v^) - pU C + 2co'fi + 2n,ior\ = 0.
i258
Thus, we obtain a homogeneous system of three linear differential equations
of the second order with constant coefficients relative to three unknown func-
tions x(t), Tl(t), and C(t).
Now, the stability analysis of the system can be carried out in the conven-
tional manner •
Let us put
where Xq, T|o, and Co ^^® certain constants*
Substituting these eij^ressions into eqs.(1.2D), we obtain a system of three
algebraic Unear homogeneous equations for determining the quantities Xq, TIo,
and Co* Equating the determinant of this system to zero, we obtain the charac-
teristic equation for determining X. On e^q^anding this equation in powers of X,
we obtain
X**.]- aX'-[-Z»X' + rX2 + ^X+/=-0,
(1.21)
282
Here and tielow, we introduce the following notations;
— (1)
/>».=-
Po
Po
c — Co + 'Cyui^;
2 r- , -
1 — e
[«o+rt,(2-e)];
^0=737 [1 +8«o'i, + 4«^ + /*.(2-e)];
B,=-
1
1 — 1
4_
1 — (
[4-(2-s)(l-v?)];
[«i, + 2«o«» + («o + «/, ) ip».] ;
c,=
1 —
K+rafcKl + '^o);
1 — e
A^
l-e
[2^,+ (l+v^)(l+4«o«J-/?l(l-vg)] ;
E^-
\ — t
2
l-e
2
- [4/io4+ 2/tfc ( 1 + V?) - 2 /7i,«o ( 1 - v^)] ;
^5=
l-e
2^0(1 -vgf
l-e
^0 = l-e'
/',=-J-[4«^-2F*.(l-v?)];
1 — e
1— e
(1.22)
(1.23)
(1.24)
/259
283
The dimensionless coefficients of danping ho (o^ ^^ elastic iDase) and n^
(blade) are deternaned "by means of
^0==^
P^
n, = ^
Po
(1.25)
The dimensionless coefficient e is obtained from the fonnula:
It is easy to explain the mechanical meaning of this irrportant coefficient.
The quantities Sy.^j and ly. h can be written in the form
/I
where p^ = V — ^*^ is the radius of inertia of the blade relative to the drag
m
b
hinge. Therefore, eq.(1.26) can be rewritten as
^=i^ri^(fr- (1-27)
The quantity pc/pi depends ipon the law of mass distribution over the blade
length and, for different blades, lies within the narrow limits of Pc/Pi ~
«^ 0.8 - 0.9.
Consequently, it can be assumed in first approximation that the quantity e
is proportional to the ratio of the total blade mass to the total system mass
(mass of the elastic base casing plus mass of the blades) and thus can be called
the relative rotor mass.
A detailed analysis of the characteristic equation shows that only oscilla-
tory instability is possible in the system while aperiodic instability is inpos-
sible (Ref.35). The boundaries of the zones of oscillatory instability (cor-
responding values of uo) can be found in the following manner: At the boundary
of the zone of instability there are purely harmonic (not danped and not in-
creasing) vibrations, which furnishes a piorely imaginary value of one of the _
roots of the characteristic equation (l.2l)-» Setting, in this equation, X = ip
(where p is a real quantity) and equating to zero the real and imaginary parts,
we obtain the following equations: /26O
284
Since the coefficients a, "b, c, d, e, and f are known functions of uJ [see
eqs.(1.23) and (1.2^)], we can regard eqs.^l.28)_as a system of two equations
with two unknowns p and cu. The values of oo and p, being the solution of the
system (1.28), represent the dimensionless angular velocity uo of rotor rotation
at which harmonic vibrations of the system_are possible, and the corresponding
dimensionless angular vibration frequency p .
We can solve the system (1.2B) by making use of the fact that the first
equation of the system (1.2S) is biquadratic with respect to p. Prescribing dif-
ferent values of u), we can determine p from this equation followed by calcula-
tion of the value of a certain quantity D(uo) equal to the left-hand side of the
second equation of the system (1.28) at this value of p:
Di^)^p^~bp^+dp^-f, (1.29)
From the results of this calculation, a curve for the dependence of D on uT
can be plotted. The values of x at which D vanishes will also be the bo\indaries
of the instability zone. We can de.monstrate that the values of uo, at which D >
> 0, correspond to steady motion of the system while the values of o)", at which
D < 0, correspond to unsteady motion.
Calculation of the unstable range is quite laborious and, for all practical
purposes, can be performed only on digital coirputers. Figures 3 .3 - 3.12 show
certain results of such calculations carried out by engineer V.G.Pashkin on the
digital conputer "Strela". The graphs permit determining the stability bounda-
ries and the dairping margins.
The stability of the system is determined in general by the following five
parameters: Vq, e, p^^ , no, n^. The graphs are plotted for the two most fre-
quently encountered values Vq = 0.25 and Vq = 0.3» Here, the value of p^^ = 0,
i.e., for a rotor with drag hinge danpers, is examined. Elastic elements are
absent. The effect of elastic elements will be discussed later in the text. For
each of the values of Vq there is a series of graphs corresponding to different
values of e. The_abscissa of each graph gives the values of the dimensionless
angular velocity au corresponding to the boundarie£ of the instability zone, while
the ordinate gives the dimenionless coefficient n^ of blade darrping at which
the instability zone is obtained. The graphs are constructed for different
values no of the dimensionless darrping coefficient of an elastic base.
As shown by these graphs, the width of the instability zone substantially
depends -upon the danping coefficients n^ and nQ. On an increase in dairping n^
(at fixed no) the lonstable range narrows and, at a certain critical value n£,
contracts into__a point. At a value n^ > r^^, the instability zone is absent at
all values of od. For exanple, at e = 0.02, Vq = 0.25 (see Fig. 3 .3), if no =_
= 0.06, the instability zone contracts^to a point as soon as n^ = 0.128; at n^ >
> 0.128, the system is stable for any cd (in this case, n| = O.I28).
The ratio 6. = — ~, whenever it is greater than unity, is conveniently /266
ni
"b
called the dairping margin.
The value of od at which the instability zone contra.cts to a point is called
285
Z261
RLg.3*3 Graphs for Determining Instability Boundaries
(e = 0.02; Vq = 0.25)*
nn^QM
Fig .3 .4 G-raphs for Determining InstalDility Boundaries
(e = 0,04; vo = 0.25).
286
I2h2.
0.0U
QM
0.08
O.tQ
0,1Z
0.1^
0J6
0.18
0,20
0.22
Rig .3. 5 Graphs for Determining Instability Boundaries
(e = 0.06; vo = 0.25) •
0,Z5
RrrOMZ
3 CO
Fig .3 .6 Graphs for Determining Instability Boundaries
(e = 0.08; Vq = 0.25).
2S7
Z261
0,15
0.10
0,15
0,10
0,05
C'OJO
Fig .3 .7 Graphs for Deterinirrmg Insta"bility Boundaries
(e = 0.10; vo = 0.25)*
0,Z5
0.10
0.15
0.10
0.05
Fig .3. 8 Graphs for Determining Instat)ility Boundaries
(e = 0.02; Vq = O.30).
288
I2hk
0.15
G,IQ
0,15
0,10
0.0^
nQ=Q,Ql
Fig. 3 .9 Graphs for Determining Instability Boundaries
(e = 0.04; Vq = 0.30).
Fig .3. 10 Graphs for Deterinining Instability Boundaries
(e = 0.06; Vo = 0.30).
289
nQ=OM
/265
Fig. 3*11 G-raphs for Deterndning Instability Boundaries
(e = 0.08; Vq = O.3O).
Kq^OM
Fig .3. 12 Graphs for Determining Instability Boundaries
(e = 0.10; Vo = O.30).
290
the critical value and can be calculated by means of the approximate formula:
i —Vq
Below^ we will give a physically clear elucidation of this formula.
It should be noted that an increase in the quantity n^ does not always lead
to an inprovement of stability. At low values of no (this can be traced from
the graphs), an increase in n^ may even lead to a small displacement of the
lower boundary of the instability acne toward smaller_values of u). This might
result in the appearance of instability at values of uo for which the motion was
steady at smaller n^ .
An increase in dairping n^ of the elastic base at moderate values_of n^ also
leads to an ajiprovement of stability; however, at very low values of n^ an in-
crease in no may lead to a rightward shift of the ipper boundary of the insta-
bility zone and thus to a broadening of the zone itself.
An analysis of the graphs permits the folio-wing iirportant conclusion: Whenr-
ever the quantities n^, and no are of the same order of magnitude and differ to
one or the^^other side by not more than a factor of 2 - 3, any increase in darrp-
ing n^j or Hq will result only in an increase of stability. At such values of n^
and "Ho, the greatest required dairping occurs approximately at
1 — Vo
For this quite inportant practical case, B«Ya.Zherebtsov*s simple approx-
imate formula"" can be derived, which shows that_the dairping margin is propor-
tional to the product of_the quantities n^ and no* This formula yields the
values of the product n^no at which the instability zone contracts to a point:
^*^o = ^^^^-. (1.30)
This approximate formula holds only at p^^ =0; its validity can be traced
from the graphs. At p^ ^ 0, we can use another approximate formula:
f^b^o — '
A> (1.31)
8-Vo
where the dimensionless quantity A is determined from the formula
'"" This formula will be derived in Section 3. Equation (1.31) will also be con-
structed there.
291
1 + Vo
-/■-^.(-^)
(1.32)
FigTire 3»13 shows the dependence of A on p^^ , for Vq = 0.25* The graph inr-
dicates that the required danping can "be suTDstantially reduced "by introducing /267
an elastic element in the drag hinge of the rotor* An ijiprovement in stability
of the system by an increase in p^ is illustrated also
"by the series of graphs in Fig.3.14.
7.2
0.8
0,^
y„-0.Z5
\
'^-^(Pto)
However, when introducing an elastic element into
the design of the drag hinge or when introducing so-
called elastic interblade coiplings, it is necessary to
recall that the bending moment acting on the blade root
in flight is generated both by the damper and by the
elastic element in the drag hinge. Therefore, upon in-
creasing the rigidity of the elastic element (on in-
creasing pt^ ) the moment exerted on the blade by the
elastic element (or interblade couplings) will increase
simultaneously with a decrease in the required moment
produced by the danper. The optimal value of p^^ should
be considered that value at which the bending moment
acting on the "blade in flight will be minimum, at con-
stant danping margin with respect to ground resonance.
This optimal value of p^^ depends on n^, and should b)e separately selected for
each helicopter. For more details, see Section 6.
0,1 OM OJ 0,8 ptQ
Fig. 3. 13 Effect of
Elasticity of the
Drag Hinge on Re-
quired Danping.
3. PhTsical Picture of Rotor Behavior in the Presence
of Ground Resonance
To elucidate the plr^sical picture of rotor behavior in the presence of
ground resonance, let us examine the following prob)lem:
Let the casing of the elastic base (Pig .3*1) execute harmonic vibrations /268
according to the prescribed law:
x=^XQsinpt,
(1-33)
where Xq and p are the vibration anplitude and frequency of the casing.
Let us examine forced blade vibrations during such movement of the casing.
The equation of motion (1.8) of the k-th rotor blade will take the following
form in this case:
?*+2«*e* + (pl + ^^<<^)?*=
^0
(1.34)
iy.h
■p^XQSin pi sin 1^^.
292
Fig.3«14 Graphs Illustrating the Effect of Elasticity
of the Drag HLnge (e = 0.04; Vq = O.366).
Ott
Considering that li^ = oot + k (k = 1, 2, *.., n) and representing the
n
right-hand side of this equation as two harmonics, we can write the equation in
the form
'i+'^4^+{pl+^ynu=
2/../,
/?2^0 icOS
n
— COS
n
(1-35)
This is the conventional equation of forced vibrations of a system with one
degree of freedom.
The right-hand side of eq.(l.35) represents the exciting force which, in
this case, consists of two conponents, each of which represents a load varying
by a sinple harmonic law with a frequency equal to (ua - p) or (uo + p), respec-
tively. Ely virtue of the linearity of eq.(1.35), the blade vibrations due to
each of these loads can be examined independently. The forced (steady) vibra-
tions of the blade will take place in obedience to the law:
^ftW = ?lCOS[(a) + /?)if + cpi] + S2COs[(o) + /?J/f + 92],
(1.36)
where f i, gs* ^>\f 93» ^i"® certain constants that are readily deteraiined from
eq.(1.35).
293
Thus, during vibrations of the casing of the elastic "base according to a
sinple harmonic law with a frequency p, the rotor blades will execute forced vi-
brations with two conibined frequencies (u) + p) and (oo - p) depending upon the
angular velocity oo of rotor rotation.
The most intense blade vibrations occur at resonance, when one of the ex-
citation frequencies (p + («) or (p - o)) is close to the natural vibration fre-
quency of the blade pij = Jv\ "•■ VqUO^.
Let us first examine the case of resonance when
YpX + vy=^\p-^l (1.37)
In this case, the quantity 5i in eq.(l.36) will be appreciably greater than
the quantity §2* so that we can neglect the second term in eq.(l.36). With this
'sinplification and with the condition (1.37)^ the law of motion of the blade
will have the form
S, = $csin |^(;?-o))^-?^^l, (1.38)
where
^P' , (1.39)
^0 = 7-; ; ^0-
Let us next calculate tjie force Px exerted on the casing of the elastic /269
base by the inertia of the rotor blades vibrating in this mode. For this, let
us find the displacement of the center of gravity of the blade system by means
of eq.(l.l2)* Substituting into these formulas eq.(1.3S) for %^^ and taking into
account that
n
^Q.os\{2<s^~ p)t +— k\ = Q
and
n
2] Sin [(2a)-/?)^+l^>fel===0,
easy transformations will yield the following law of motion of the center of
gravity of the blade system:
x^=-x^^^^cospt\
^^Q sin pt.
(1.40)
If we take into account that the coordinates of the center of gravity in
294
the coordinate system y:' kz' referring to the casing are e:xpressed by the formu-
las Xq = Xc - X and Zc = z©* then the center of gravity of the system of blades
in this coordinate system moves in accordance with the law:
(1.41)
Thiis, at resonance when the equality (l#37) is satisfied, the center of
gravity of the blade system describes, in the coordinate system fixed with re-
spect to the casing, a circle of radius ^ Pc?o» -^^ this case, the angular ve-
locity of its rotation with respect to this circle is equal to the frequency p
of the given vibrations of the casing.
Let us then determine the force P^ acting on the casing, by means of the
formula P^ = -nm^Xc . Here, we obtain the following expression:
P.^
■nrrtyp^
Substituting here the expression for 5o f^onit eq*(l»39)j we obtain
P^^nm^p^
1+-^
V^p2
8 lv,h^b(^~P)
Xo COS pi.
(1.42)
Thus, during vibrations of the casing by the harmonic law [eq.(1.33)] and
under the condition of blade resonance [eq.(1.37)]> the force exerted on the
casing by the vibrating blades varies in time by a harmonic law with the same
frequency p, with a vibration phase tt/2 (with respect to the vibrations of the
casing) and is proportional to the azimuth Xq of the vibrations of the casing.
Equation (1.42) can also be represented in the form
P^==-nm,x + nm^p^-^
v?p2
8 lyj, .n^(o>— ;?)
Xf) COS pi.
On deriving the equation of the casing (1.2) under the effect of the /270
force Px given by such an expression, we obtain
niQX + 2kx -}- cx = — nm^x ~\- nmf,p'^ ~
^Ip'^
■Xq cos pi.
8 Uh nyitsi—p)
Using our previously adopted notations, this equation can be written as
295
x + 2n^ + plx=-ff^ f. ^'f x,cos pt. (1,^3)
If it were possible to find the parameters of the system at which the law
of motion of the casing Ceq.(l#33)] satisfies this equation, this woiild mean
that, at such system parameters, purely harmonic motion (undanped vibrations)
with a frequency p would be possible • Substituting eq.(1.33) into eq#(1.43)> it
is easy to demonstrate that this is obtained when the following two conditions
are satisfied:
P = Pq\
rirji
o'^b-
^Pq
8 (o) — /?o)
(1.A4)
Furthermore, it should be recalled that eq#(l#43) was derived from the con-
dition of blade resonance, i.e., under the condition (1.37) which, taking Po = p
into account, can be written in the form
Yp\ + ^y^\Po--^V (1.45)
From this equation, one can determine the value of the critical angular ve-
locity oJcj. of rotor rotation at which undanped vibrations in the system are pos-
sible .
Equation (1-44) gives the value of the product n^n^ at which undanped vi-
brations are possible; then, as now is obvious, this foimula together with the
condition (1.45) will yield the approximate formula (I.3I).
Thus, undanped vibrations are possible only at a value of u) at which two
resonances occur simultaneously: resonance of the blade [condition (1.45)] and
resonance of the elastic base p = Po» At such a value of 00 and on satisfying
the condition (l./|4), the natural vibrations of the rotor on an elastic base can
be sustained by a variable exciting force generated by the vibrating blades,
which here are in a state of resonance.
A study of eq.(1.35) shows that blade resonance is possible in two cases,
namely: when one of the combined frequencies (p + cjd) or (p - uu) coincides with
the natural frequency of blade vibrations, i.e.,
and when
Of these two cases, we examined only the first. For the second case, all
derived formulas are obtained in the same manner except that, in all egressions,
the quantity od is replaced by the quantity -cu, including also in eqs.(l.Zt4).
This e^q^ression shows that, atp^=|p+aol, undanped vibrations are possible
296
I I I ■■■■■
only if noH^ < 0, meaning that in this case one of the quantities no and n^
should "be negative. Consequently, ground resonance is possitile only at p^, =
= |p - 0)] and inpossible at p^ = jp +00].
Let us next peruse the resonance diagram (Pig.3.15). This diagram gives /271
the curve of the nat^aral blade frequency p^ as a function of the angular ve-
locity 0), TO-th stperposition of the straight
lines p = Po + t« and p = |po - ca)] . The diagram
^l / is plotted for the case of p^^ < po •
As we see from the diagram, there are two
values of oo at which the condition p^ = |po - ^\ ,
corresponding to the points A and B, is satis-
fied • For the point A, we have the condition
Pb = Po " ^ and, for the point B, the condi-
tion p^ = OJ - po •
Thus, in the first case ud < po and in the
Fig .3. 15 Resonance Diagram. second, uo > po • Turning to the second condi-
tion of the system (l.Z^), we see that it can
be satisfied (at positive values of Uq and n^)
only for co > po» Consequently, of the two possible values of ud at which blade
resonance is possible, only one (o) > po) can correspond to undanped vibrations
of the system.
Let us determine this value of o) and call it {is)^^ ) critical. Solving
eq.(l.45) relative to uo and discarding one of the obtained values (uo < PoJ* we
find
^cr=P0
(1.46)
At pi3 =0 we obtain the formula
^ Po
1 — Vq *
(1.47)
Substituting the value of uOcr from eq.(1.46) into the second condition of
the system (1.^4), we find
7? w __£( ! — Vq) ,
where
I + vq
Vo +
/-^.(■^l
297
These formulas exactly coincide "with the approximate equations (1-31) and
(1.32).
The reasonings set forth here, together with the stability analysis given
e(l - Vn)
in Section 2, permit to state: The condition r^n^ > A always pro-
vides stability at the critical angular velocity of rotor rotation determined ty
eq#(l#46). However, as indicated in the analysis of^the graphs in Figs .3 '3 to
3 .12, this condition holds only when the quantities n^ and no are of the same
order of magnitude. This means that ensurance of stability at oa = cju^j. does not
definitely ensure stability at any ud.
4. Rotor on an Isotropic Elastic Base
The theory of stability of a rotor on an elastic base presented in this
Section holds only if the ntomber of rotor blades n > 3 and if the elastic base
has only one degree of freedom, namely motion along the Ox-axis (Fig .3. 2). /272
However, an analogous stability theory can be constructed also for the more
general case where the elastic base has two degrees of freedom: displacement
along the axes Ox and Oz. A stability analysis for this more coirplex system is
rather cumbersome. On the other hand, in practical application one can almost
always use the formulas for the case of an elastic base with one degree of free-
dom. Thus, this can be done whenever the natural longitudinal and lateral vibra-
tion frequencies of the helicopter on an elastic landing gear (see Sect. 5) are
far apart.
It is of interest to give a few siirple results, obtained in the stability
theory for a rotor on an elastic sipport with two degrees of freedom in the
special case of a so-called isotropic elastic support when the stiffness and
danping of the elastic attachment of the casing to the base are identical in both
directions (Ox and Oz). In this case, the elastic and danping properties of the
base are identical in all directions parallel to the plane xOz. Therefore, such
a base or sipport is called isotropic.
Let the stiffness and danping of the isotropic base, identical in directions
of the Ox- and Oz-axes, be characterized respectively by the coefficients c and
k, so that the forces P^ ^^^ ^z applied to the base are related with the cor-
responding displacements x and z by the formulas
^ dt '
r» dz
Pz=^ —CZ~K -,
dt
(1.4s)
It is found that, in this case, there can also be instability of the rotor
on an elastic base. Here the unstable range is close to the same value of o) =
= UD^^ as before:
298
i + l /vg+P.,(i-v§) (1.49)
i-v;^
At Pto = 0, just as before, we obtain a sinpler formula:
'""•""rr^o" (1.50)
In this case, the quantities po, Pbo* ^"^ "^o ^^® detemiined, as usual, by
the formulas:
w^
(1.51)
Analogous formulas are obtained for detennining the required dairping, but
the required danping in this case is greater by a factor of 2.
The formula for the required dairying at which the instability zone con- /273
tracts to a point, has the form
^o«b = '-^^A. (1.52)
The quantities e and A are determined, as before, by eqs.(l.26) and (1.32) •
Section 2. later al V ibrations of a Single-Rotor Helicopter
1, Preluninary Comments
In calculating the vibrations of a heUcqpter on an elastic landing gear we
can regard the fuselage as a perfectly soUd body attached to a stationary base
(ground) by means of a system of elastic elements.
The calculation of ground resonance of a helicopter, as will be shown below,
can be reduced to the calculation of a rotor on an elastic base, examined in
Section 1. The initial data for such a calculation (characteristics of the
elastic base) are derived from a preliminary calculation of natural vibrations
of a rigid fuselage on an elastic landing gear.
A helicopter regarded as a solid body on an elastic landing gear has six
299
degrees of freedom. However, since the fuselage, as a rule, has a plane of syror-
metry, the longitudinal and lateral natural vibrations of the helicopter can tie
examined independently of each other.
For a single-rotor helicopter -with an elongated fuselage, the lateral vi-
brations are generally calculated from the viewpoint of ground resonance. In
the presence of longitudinal vibrations, the danping margin for eliminating
ground resonance is appreciably greater. Therefore, to calculate ground reso-
nance of a single-rotor helicopter it suffices to examine only lateral vibrations
(see also Sect. 5) •
When examining the lateral vibrations, we must take into account three de-
grees of freedom:
1) lateral displacement of the center of gravity of the helicopter;
2) rotation of the helicopter about the longitudinal axis (rolling);
3) rotation of the helicopter about the vertical axis (yawing).
Generally speaking, the helicopter vibrations corresponding to these three
degrees of freedom cannot be regarded as independent. For exanple, on lateral
displacement of the center of gravity of a helicopter, forces are generated that
cause rolling, etc.
However, in a single-rotor helicopter for which the longitudinal fuselage
dimensions are relatively large in conparison with its lateral dimensions (this
need not be the case, e.g., for helicopters of coaxial and side-by-side con-
figurations), the yawing vibrations are weakly related with lateral vibrations
of the helicopter and with its rotation about the longitudinal axis. Therefore,
in first approximation, the yawing vibrations for a single-rotor helicopter can
be regarded as independent. Furthermore, during yawing vibrations of a heli-
copter the displacements of the center of the rotor in the plane of rotation are
relatively small (in conparison with lateral vibrations) so that, as a rule,
yawing vibrations for a single-rotor helicopter are not dangerous so far as
ground resonance is concerned. As we will see later (Sect.5)> such vibrations
are dangerous for helicopters of fore-and-aft and side-by-side configurations.
Thus, in studying the lateral vibrations of a single-rotor helicopter it is
sufficient, in first approximation, to consider the fuselage as a body with two
degrees of freedom:
1) lateral displacement of the center of gravity of the helicopter;
2) rotation of the helicopter about the longitudinal axis (rolling).
Mth such sinplifications, the problem of natural lateral vibrations of /274
a helicopter can be reduced to the problem of nattiral vibrations of a two-dimen-
sional solid body elastically attached in its own plane (Fig.3.16).
2. Lateral and Angular Stiffness of landing Gear .
Flexural Center
Let a rigid body A, simulating a helicopter fuselage, be mounted to a sta-
tionary base by means of a system of springs (Fig.3»l6). We select a fixed co-
ordinate system ycoZ, directing the axis Coy along the axis of symmetry of the
300
"body and the axis QqZ along the axis of the horizontal springs c^»
If, to the "body A, a force P^ parallel to the axis Cqz at a distance y from
the point Cq is applied, then the deformations of the springs will cause the
"body A to "be displaced in its own plane so that its axis of symm'etry will come
to occTj^jy a certain position Coy' .
Let us denote "by cp the angle of rota-
ly' tion of the axis of symmetry of the
body (angle of roll) and "by 2 the dis-
placement of the point Cq (segment
Let the springs have linear char-
acteristics • Then, as is known, a
point of application of force is al-
ways found on the axis Coy (or a value
of y) at which the angular displace-
ment cp of the "body will be equal to
zero, meaning that, upon application
of the force P^ at this point, the
body will undergo purely forward dis-
placement (cp =0). We will call such
a point the flexural center of the
shock absorber system.
If, to the body A, a couple with
a moment M is applied, then the body
will undergo only angular displace-
^^^'
Fig .3. 16 Diagram of Elastic Mount-
ing of Helicopter.
ment - turning about the flexural center.
It is easy to see that, for the siirplest shock absorber system, as it is
shown in Fig .3. 16, the center of gravity will be located at the point Cq* The
position of the center of gravity of the shock absorber system is conveniently
characterized by the magnitude of the distance e from the center of gravity c of
the body to the center of gravity Cq •
If, to the body, a force Py directed along the axis of symmetry Coy is ap-
plied, then the body will undergo only forward displacement y along the axis Coy<
Since the characteristics of all elastic elements of a shock absorber system are
linear, the forces P^, Py and the moment M of the cot^^le are linearly related
with the corresponding displacements y, 2, and cp of the body A.
Let this relation be expressed by the formulas:
(2.1)
(2.2)
(2.3)
We will call the (quantities Cy, c^, and c^, respectively, the coefficients
of vertical, lateral, and angiolar stiffness of the shock absorber system.
301
The elastic properties of the shock alDsor"ber leg are fully determined by
foior parameters: position of the flexural center (e) and coefficients of stiff-
ness Cy, 0^, and c^p.
For the sinplest shock al3sor"ber system depicted in Fig.3»l6, the coef- /275
ficients of stiffness of the shock alDsoiption can te determined "by means of the
formulas :
(2.4)
where
Cy and Cg = coefficients of stiffness of the vertical and horizontal
springs ;
2a = distance between the axes of the vertical springs (wheel
track) .
Fig .3 .17 Various landing Gear Config-urations-
a - Pyramidal; b - With vertical struts.
The types of helicopter landing gears are mainly of two variants:
1) pyramidal landing gear;
2) landing gear with vertical struts.
The elastic shock absorber systems corresponding to these two types of land-
ing gear are depicted in Fig .3. 17, a and b.
The pneumatic tires in this scheme can be considered perfectly rigid, a.id
their elasticity can be simulated by special springs with stiffnesses c^"" and
302
tire-
equal, respectively, to the vertical and lateral stiffness of the pnetunatic
The coefficient of vertical stiffness of the tire can be determined from
the diagram of static tire coirpression, which is always available in the catalog
of wheels and represents the ratio of the magnitude of the force conpressing the
tire toward the rim siorface to the magnitude of the corresponding tire coirpres-
sion» The lateral stiffness of the tire, if there are no data available, can
also be determined e:;^erimentally* The magnitude of lateral stiffness of the
tire must also be known for calculations of shimmy. Therefore, if shimmy has
been calculated for a given wheel, the magnitude of the lateral stiffness will be
known. For an approximate determination of lateral stiffness of a tire we can
also use Table 3 •I*
TABLE 3.1
Type of Pneumatic
Tire
Arched
Semi- balloon
High-pressure
Pff/rfi^
cric^
0.7—0.9
0.4—0.64
0.3—0.4
The shock absorber strut of the landing /276
gear in Fig.3*l7>a a-nd b is also replaced by a
certain spring of stiffness Cg . a • ^^ reality,
the shock strut of the landing gear is a non-
linear elastic element, and its characteristic is
determined by the diagram of static coup res si on
of the strut which gives the force P acting on
the strut as a function of the stroke s of the
strut .
In calculating small vibrations, the strut
can be replaced by an equivalent linear elastic
element (spring) whose stiffness is determined
by the formula:
dP
ds
(2.5)
where Sst is the standing conpression of the shock absorber.
In a landing gear system with vertical struts ( Fig .3. 17, b) the flexural
center of shock absorption is always situated at the point Co on the ground sur-
face. The coefficients of stiffness of such a landing gear are determined by
eqs.(2.4) where c^ and Cy are equal, respectively, to
-cr;
nP'^
'y-'
Cs..^cP-
(2.6)
For a pyramidal landing gear (Flg.3*41ja), the flexural center is always
above the ground surface, and its position must be calculated by special formu-
las which we will give below.
The pyramidal landing gear is a special version of a more conrolex landing
gear system developed by the British Bristol Aeroplane Co. (Ref .39) and depicted
303
(schematically) in Flg*3.18. This landing gear system differs from the pyrami-
dal landing gear by the presence of a rocker AB and a special horizontal spring
of stiffness Csp . In this system, the height of the position of the flexural
center Cq (i»e., the quantity e) can "be varied ty selecting a certain spring con-
stant Csp . In particular, by choosing a certain value of Cgp it is possible to
obtain a position at which the flexural center of the shock absorber system co-
incides with the center of gravity of the helicopter. In this case, as will be
seen later, there is no coupling between the rolling vibrations and the lateral
vibrations of the helicopter, which permits obtaining good helicopter character-
istics with respect to ground resonance (see Sect .4, Subsect. 3).
For the landing gear system depicted, in Pig. 3. IS, we can write the follow-
ing formulas which can be derived easily by the usual methods of structural
mechanics:
1
c,=
1
2 UA ^ c^a^} " (2.9)
^9 f h I ^0
Ml
where
ho = distance between ground surface and the point F of intersection of
the axes of the shock struts (see Fig .3. 18);
I = distance between shock absorber axis and the point A;
t^ = distance between the points F and A;
1 2 = distance between the point F and helicopter center of gravity.
As a special case, the derived formulas contain the formulas for calcu-
lating the pyramidal landing gear (see Fig.3.17,a). To obtain formulas of the
pyramidal landing gear, it is necessary to set Cgp = » in eqs-(2#7) and (2.8).
3* Natural lateral Vibrations of a Helicopter
Let us now tiorn to Fig .3. 16. In studying the lateral vibrations, let us
use, for the body A, two degrees of freedom corresponding to the coordin3.tes cp
and z. We will impose an additional limitation on the motion of the body A: We
will stipulate that the point 0^^ belonging tc5 the body A at a distance a^ from
the center of gravity of the body remains stationary. Then, the body A will have
one degree of freedom - rotation about the point (\ . The equation of natural vi-
brations of the body A, attached in this manner, will have the form
V+^o,?=-0, (2.11)
304
where
Iqj^ = moment of inertia of the "body relative to the point Oi^ :
(2.12)
I« = moment of inertia of the "body relative to the center of gravity;
mass of the "body*
-c
m
Fig .3. 18 landing Gear Scheme of the
Bristol 192 Helicopter.
The coefficient Oq^ represents
the angular stiffness of the shock
absor*ber system upon rotation of /278
the body A relative to the point Oj^ .
The quantity Cqj^ is readily deter-
mined if the position of the flex-
ural center Cq of the shock ab-
sorber and its angular stiffness Cqo
and lateral stiffness c^ are
knovm.
Upon rotation of the body
through an angle cp relative to the
point Ojj, the flexural center is
displaced by the amount
e=(p(a^— e).
(2.13)
In this case, a force P^ = c^z
directed to the left and a coiiple
of moment M = c^jcp directed counter-
clockwise will be applied to the
body at the flexural center. The
moment of these forces relative to
the point 0^ is
Hence, we obtain the following formula for the angular stiffness Cqj^ :
The natural vibration frequency of the body with the fixed point 0^. is
or
Pk--
(2.15)
305
During vilDrations of the system at the point Oj^ , a reaction force R arises
which will depend on the position of the point Oj^ . If we could select a point
of attachment Oj^ (a value of a^) for which R = 0, this would mean that such a
point 0^ is the natural vibration node of a free system with a movable point Oi^ ,
and the corresponding frequency p^ is the natirral vibration frequency of a free
system*
The reaction force R is readily determined: During vibration, the body A
is loaded by the inertia force Fjjj applied at the center of gravity and parallel
to the axis 0^,
and also by the coiple of inertia forces. The forces exerted by the shock ab-
sorption on the body are also reduced to the horizontal force
and to the couple. Therefore, projecting all forces acting on the body onto the
axis CqZj we obtain
If
then
^— [^z {^k — ^) — ^Pl<^k\ 9o ^os p^t.
Equating this expression to zero yields
PVom this, we obtain the following formula relating the natural frequency /279
of the system with the position ai^ of the vibration, node:
(2.16)
flft = -
l-lM'
where
Pl=^- (2.17)
m
Excluding the quantity p^^ from eqs.(2.15) and (2.16), we obtain a quadratic
equation for determining the quantity a^ . This quadratic equation always has
two real roots a^ and ag, which correspond to the two natural vibration over-
tones of the system. For each overtone, we obtain a certain natural vibration
frequency p^ which, at a known a^, can be determined from eqs.(2.l6) or (2.15) •
To determine the natural vibration frequencies pi^ and the corresponding
306
12m.
2.0
10
^
—
-^
-^ 3t
ihiiii!
^
^
^^
>f-C
rv/
^^"
\ IS
1
0.5'/
i.O'
/,0
2.0
Flg.3«19 G-raphs for Determining
the Natural Helicopter
Frequencies.
Fig .3 •20 Graphs for Determining
the Position of Vibration
Nodes •
pj- Z/5cyc/m/n
Jo/ first overtone
Vibration node
of second overtone
V//-;r;777/yr/////r/r////7/ ' 7/7777777777777^7^77777777777777
Oft Vibration node
Fig .3. 21 Characteristic Vibration Modes of the
First and Second Overtones.
Fig .3. 22 Diagram of
a linear Elastic Ele-
ment with Dajiping.
307
quantities aj^, it is convenient to reduce all formulas to a dimensionless form,
introducing the notations:
a,=^; (2.18)
e
-p,=M.; (2.19)
Pz
Xz
_ Ic . (2.20)
(2.21)
The final formulas for deterndning a^ (k = 1, 2) and p^ (k = 1, 2) can be
written, in such notations, in the form
%2 ="- y^ ± V'f+^, ( 2 . 22)
where
i + P_^ _ (2.23)
^ 2 '
?* = ]/l-^; (^f A=l,2). (2.2^)
For convenience of calculating the positions of the natural vi"bration nodes
of the first and second harmonics and the corresponding vibration frequencies.
Figs. 3. 19 and 3.20 show graphs calculated by means of eqs. (2. 22), (2.23), and
(2.2^).
The lower of the frequencies p^ and pg vri_ll "be called the frequency of the
first vibration overtone while the higher frequency will be that of the second
overtone. The vibration node of the first overtone is always below the center
of gravity of the helicopter (a^ > O) while the vibration node of the second
overtone is always above the center of gravity (ag < O) .
Figure 3 ,21 shows the characteristic vibration modes of the first and second
harmonics for a single- rotor helicopter with a pyramidal landing gear.
4. Determination of Damping Coefficients
The danping of vibrations (i.e., absorption of energy -during' vibrations) is
generally small and can be neglected in determinations of the natural frequencies
and positions of the nodes (as was done in Subsect.3)-
Danping of vibrations takes place mainly in the shock struts of the land- /281
ing gear. Danping in pne\:imatic tires can be disregarded in first approximation.
Let us examine the system depicted in Fig .3 •16. Let certain linear elastic
elements with danping be installed in place of the springs. Such an element is
schematically shown in Fig #3. 22. let the force P acting on this element and its
308
displacement s (stroke of the element) "be connected "by the relation
P=cs + k^. (2.25)
We will call the quantities c and k, respectively, the coefficients of
stiffness and danping of the elastic element.
We will denote the coefficients of stiffness and damping of the elastic
elements in the system shown in Fig.3*l6 loy c^, Cy, k^, and ky, respectively"^'*.
The equation of vibrations of the tody A relative to the node can be written
analogously to eq.(2.1l) in the form
^ojp + >^o^? + Co^9 = 0, ( 2 . 26 )
where the quantities Iq^ and Cqj^ are determined from eqs.(2.l2) and (2.14) and
the quantity k^j^, by the formula
ko,=^2kla^^2k'^(a,-ey. (2-27)
We will call this quantity the angular coefficient of dairping of the shock
absorber system upon rotation of the body relative to the -vibration node 0^^ .
Equation (2.26) can be written in the form
? + 2/i;,? + /?2^=-0, (2.2S)
where p^^ (k = 1, 2) is the frequency of the k-th vibration overtone, while the
dairping coefficient n^ is determined by the formula
'~^,^ (2.29)
The natural vibrations of the k-th overtone of the helicopter can be de-
scribed approximately by the law:
<p_cpo^-V cos(/7^^+ip), (2.30)
where
cpo = initial angle of deflection;
ijr = phase angle.
The natioral vibration frequency p^ can be taken as approximately equal to
the natxiral vibration frequency of the k-th overtone, calculated without con-
""'^ The manner of determining the coefficient kg and ky will be shown in Subsec-
tion 5 of this Section, and also in Subsections 1 and 2 of Section 3.
309
sideration of darping.
In order to calculate the quantities ky and k^ for a specific landing gear
system (see ?i-g.3*17,a and "b) it is necessary to determine first the effect of
the system formed "by the shock strut and the tire#
5 • Combined Action_of the _Syst em. 5hock _ &fcrut-Pneijmatic Tire
We -will discuss here a landing gear system -with vertical struts (see
RLg.3«l7,'b). The tire-oleo ccmbination represents tiwo springs with stiffnesses
Cs.a ^^ ^pn connected in sequence.
Let us examine the work done "by such a system for the case in which the /282
shock absorber has darping. Such a system is shown in Pig. 3 •23* Let the shock
absorber have a linear characteristic analogous to eq.(2«25)j
^.a — ^f.> -^sa ~r "*a ~T~ *
at
(2.31)
After deriving the equations of motion of the tire-oleo system, it is easy
to show that, with the given harmonic law of
variation of its total stroke s with a fre-
quency p, the force P acting on the shock ab-
sorber is expressed by the formula
P=c^
^^^ + ^e^ 77'
(2.32)
Fig .3. 23 Schematic Diagram of
Tire^leo System.
where s^^ and k^q are the characteristics of
some equivalently Unear shock absorber of
the conventional type (Fig. 3 .22) and can be
determined by means of the formulas:
(2.33)
(2.34)
Thus, in vibration analysis a landing gear system with vertical struts (see
Fig .3. 17, b) can be replaced by the system shown in Fig .3. 16, in which the char-
acteristics of elasticity and dairping of vertical springs are selected accord-
ing to eqs.(2*33) and (2.34) • At kg. a = 0, eq.(2.33) yields a value of c^.
equal to the value of Cy obtained by the second equation of the system (2.o),
Consequently, in the presence of dairping, eq.(2.6) - generally speaking - does
not hold. However, for an approximate calculation of natural frequencies we can
use eq.(2*6) for determining Cy, since the value of c^^ determined by eq.(2.33)
310
is close to the value of Cy found from eq,(2«6). After determining the natural
frequency p, the value of Cy can fee refined l>y eq.(2.33), followed by refine-
ment of the calculation of the frequency p •
For an exact calculation of the natural frequencies we can use the method
of successive approximations (in practice, the above correction equivalent to
the first approximation is sufficient) or else the following method: Prescrib-
ing the values of Cy in the interval
"s-a ^ pn
.+0'
pn
we find the natural frequencies and then, from eq,(2.33)> we find for the given
c^q = Cy the corresponding value of kg
As a result of this calculation, the
graph of the natural vibration frequencies of the system can be plotted as a
function of kg, a • Calculations show that the natioral vibration frequencies and
modes depend little on the quantity kg^^ • Therefore, in practical application
it is sufficient to carry out the above-described approximate calculation with a
subsequent single refinement of the frequencies.
To calculate the danping coefficient i\ [eq.(2»29)] we can set, neglect- /2B3
ing the tire danping, in eqs#(2.27)-
For a pyramidal landing gear, the danping of vibrations can be calculated
approximately by the same method; however, in calculating kg^, the quantities
Cs.a and kg. a in eq«(2.34) must be substituted
by the values of the so-called stiffness and
danping of the shock absorber reduced to the
red
Fig •3. 2^ Equivalent Danping
as a Function of Shock
Absorber Danping.
tire c_ _
formulas
and k«
determined by means of the
(2.35)
where I is the distance between the shock ab-
sorber axis and the point A (see Figs. 3-18 and
3. 17, a) of the intersection of the axes of the
lower inclined struts.
Let us discuss in more detail the dependence of the equivalent danping co-
efficient koq of the tire-oleo system on the quantity kg,a . Plgure 3»2U gives
the graph of this dependence. As indicated there, the quantity k^^ increases
with increasing kg^^ only up to a certain value k^.^ - ^s?i at which maximum
danping k^q^ is attained. Upon further increase in kg^^, the danping of the
tire-oleo system decreases .
311
From eq.(2*34) it is easy to obtain the expression for the cptimal value
of kl^l:
j^oDt— ^|/»+^J.a
P
(2-36)
For this value of kg.a* eqs.(2.33) and (2.34) give the corresponding values
of c2q and k^q"" :
'"'T^'+lzf^} (2-37)
1 C^n
'^'-■^■P^^- <-^«'
We see from the last formula that the maximimi obtainable value of k^^^ is
greater the smaller the ratio — t^-^ and the larger Cpj^ . Therefore, from the
c^
pn.
viev/point of danping of lateral helicopter vibrations, the tire should be as
rigid as possible and the shock absorber should be as little rigid as possible.
At inproper selection of the landing gear characteristics (greater relative
stiffness of the shock absorber — ^-^-^ ) it may happen that ground resonance is
iirpossible to eliminate no matter how far the shock absorber dairping is in-
creased.
6. Reduction of the Problem to Calculation of a Rotor /284
on an Elastic Base
After calculating the natural vibrations of the helicopter on the ground,
i.e., after determining frequencies, position of the vibration nodes, and danp-
ing coefficients for both natural vibration overtones, it becomes possible to
calculate ground resonance in first approximation by reducing the problem to
calculation of a rotor on an elastic base.
It would be possible to carry out an exact calculation of ground resonance
by deriving the equations of motion of the rotor blades and of the helicopter
body in a manner similar to that used in Section 1 for a rotor on an elastic
base. Then, the order of the characteristic equation would be higher the more
degrees of freedom of the helicopter on an elastic landing gear are taken into
accoimt. However it would be necessary in each case to perform cumbersome cal-
culations .
An approximate calculation based on reducing the problem to a rotor on a
flexible support permits using established data obtained for a rotor on an elas-
312
tic base. The accuracy of such a calculation is adequate for practical applica-
tion.
The essence of such an approximate calculation is as follows: An individual
calculation of ground resonance is carried out for each natural vibration har-
monic of the helicopter on the ground; in this, the helicopter casing is conr-
sidered as a body with one degree of freedom - rotation about the corresponding
vibration node.
The equation of motion of a helicopter with a fixed vibration node has the
form
^oJ^+^o,^ + Co,9-'-=Pih+aj,). (2-39)
The right-hand side of this equation represents the moment of the force P
due to the vibrating rotor blades relative to the vibration node of the harmonic
in question. The quantity h is the distance from the plane of the rotor to the
center of gravity of the helicopter.
We then introduce a new variable x = cp(h + a^) representing the degree of
displacement of the rotor center. Equation (2»39) can now be rewritten in a
form analogous to the equation of motion [eq.(1.2)] of an elastic base:
where the quantities m^q , k^q, c^q represent the mass, danping, and stiffness of
the equivalent elastic base and are calculated by means of the formulas:
;,,. ^ /q, ^ Ic + rnal ^ (2.41)
%~ia,-^h)2' (2.43)
Thus, the problem reduces to the calculation of a rotor on an equivalent elastic
base whose characteristic is determined from eqs.(2«4l)» (2-42), and (2.43) •
It is easy to demonstrate that, for calculating ground resonance by means
of the formulas given in Section 1, we require only three characteristics of the
elastic base mo = m^q, no = n^, [see eq.(2-29)], and po = Pk, which are obtained
from calculations of the natioral lateral vibrations of the helicopter.
Thus, for each natural vibration overtone of the helicopter on an elastic /2B5
landing gear, we carry out an approximate calculation of ground resonance by
means of the formulas derived for a rotor on a flexible sijpporfc (Sect.l). In
such a calculation, we can determine the boiindaries of the instability zones and
313
the magnitudes of the danping coefficients of the "blade and landing gear, which
are required for eliininating instability vath respect to each vi"bration over-
tone •
?• Analysis of the Results ofG;round,_ Resonance Calculations
The results of ground resonance calculations are conveniently represented
as a diagram of safe rpm. Figure 3*25 shows such a diagram for the Mi-4 heli-
copter. The alDscissa gives the rotor ipm while the ordinate shows the rotor
thrust T.
The vibration frequencies of the helicopter on the ground are calculated in
two variants:
1) shock struts of the landing gear operative;
2) shock struts inoperative.
Tkg
80QQ
SOOO
mo
ZQOQ
Thrust with extreme \ -^
left correction -^
Thrust with"
extreme
right
correction
Z50 a rpm
F±g*3»25 Diagram of Safe Revolutions for the Mi-4
Helicopter,
v^ - Frequency of first vibration overtone with struts inopera-
tive; Vq - Fre<^ency of second vibration overtone with struts
inoperative; v^" and vf - Frequency of first and second overtones
with struts operative.
This must be done since the shock struts of the landing gear operate only
when the coiipressive force of the strut is greater than the so-called force of
pretightening of the shock absorber. Therefore, at a certain (critical) value
of thrust T = Tqj. of the rotor, the force coirpressing the strut becomes less than
the force of pretightening of the shock absorber and the strut ceases to operate.
At T > Tcr, the shock absorber struts behave as rigid rods, and the helicopter
is able to rock only as a consequence of elasticity of the tires which are vir-
tually devoid of danping. The unstable range of the helicopter with inoperative
struts .is usually inpossible to eliminate and is always present in the diagram
of safe Ipm (this range is hatched in Fig.3.25)«
The boundaries of the instability zones and the zone of possible values of
rotor thrust T and ipm n peiroitted by the rotor and engine control systems
31^
(system pitch-gas) are plotted in the diagram of safe rpm. If none of the pos-
sible combinations of the values of T and n come to lie outside the boundaries
of the unstable range, stability of the helicopter is ensured. In this case, it
is always desirable (for greater details, see Sect .6) to have a certain stability
margin, i.e., sufficient distances in the diagram between the boundaries of the
unstable range and the boundaries of the possible T and n values.
For a single-rotor helicopter with conventional landing-gear design /2B6
(pyramidal landing gear or gear with vertical struts; see Fig.3.l7>a and b), the
frequency of the first vibration overtone generally is below the operating rpm
of the rotor (Fig.3»25), whereas the frequency of the second overtone usually is
above this rpm. Therefore, selection of the danping coefficients must ensure
absence of an instability zone of the first vibration overtone with the struts
operative. Here, a reliable danping margin is required. The stability margin
with respect to the second vibration overtone can be ensured in practice only
"with respect to rotor rpm" and can be characterized by a certain quantity T|:
^=-^^, (2.46)
where
n^ax = maximum possible rotor rpm;
n^ = rpm corresponding to the lower boundary of the instability zone
of the second overtone.
Section 3» Characteristics of Damping of landing Gear and Blade .
]jrif luence on G-round Resonance
1. Det ermination of the Damping Coefficient of the landing
Gear Shock Absorber
In calculating the natural frequencies of a helicopter we assumed that the
shock absorbers of the landing gear have linear characteristics . Actually, the
characteristics of the shock strut of a landing gear are nonlinear as a rule.
However, for calculating small helicopter vibrations (as usually done in the
theory of nonlinear vibrations) the nonlinear shock absorber can be replaced by
some equivalent linear shock absorber, for which the coefficients of stiffness
and danping depend on the vibration frequency and anpHtude. For an approximate
determination of the stiffness of an equivalent linear shock absorber we have
proposed eq.(2-5). To determine the coefficient of dajiping k of an equivalent
linear shock absorber, we can suggest another sinple formula. This formula can
be derived if we consider as equivalent a linear shock absorber which, per vibra-
tion period, absorbs the same energy as a real shock absorber at the same vibra-
tion frequency and airplitude.
The most common designs of shock struts of a given landing gear absorb
energy because of friction in the packing glands and of the hydrauHc resistance
set vp when the hydraulic fluid is forced through small orifices.
If we assume the force of hydraulic resistance in such a shock strut as pro-
315
p03?tional to the square of velocity, then the dependence of the force of resist-
ds
ance P of the strut on the rate of its conpression can "be e^ipressed as
dt
P=-
where Pq is the force of friction in the gland, while a^ and a^ are the coeffi-
cients of hydraxiHc resistance of the strut in the forward and return strokes •
Let the rod of the shock absorber execute vibrations according to the law
• ds
s = So sin pt and, consequently, s = = ps© cos pt.
dt
let us then calculate the energy absorbed by the shock absorber under /2&1
these conditions during one oscillatory period. This energy is determined by
means of the formula
On calculating this integral for the case in which the function P(t) is
given ty eqs.(3»l)> we obtain
where
2 * (3-3)
ds
The danper with linear danping fp = k ^j, under the same conditions, ab-
^ dt ^
sorbs the energy A^ = nkpso during one oscillatory period.
A coirparison of the expressions for A and A^ yields the following formula
for determining the coefficient of an equivalent Unear danper:
Thus, in a real shock absorber strut, the quantity k^^ depends on the anpli-
tude So and on the vibration frequency p, a fact that must be taken into con-
sideration in calculating helicopter vibrations.
316
Figtire 3*26 shows the quantity k^^ as a function of the vibration anpli-
tude So* On an increase in vibration anplltude, the quantity k^tj decreases,
reaches a iranimum value k^^^ at a certain anplitude Sq and, \:pon a further in-
crease in anplitude, rises again.
An analysis of eq.(3«4) readily yields the following formulas for detenninr-
ing the minim'um value k"^^ and the corresponding vibration airplitude s'o of the
rod:
3_Po
2 o
(3.5)
(3.6)
We see from eq.(3»5) that the minimum danping of the shock absorber does
not depend on the vibration frequency and amplitude. For a rough estimate of
the danping capability of the shock absorber system of the landing gear it is
convenient to use eq.(3»5) and to assume, in calculating the tire-oleo system,
that kg^ a = ^eq^ • Here, it is also useful to determine the quantity s'o by means
of eq.(3.6) .
^ h
Whenever there is an occasion to make dairping tests on full-scale shock
struts, it is suggested to perform such tests since the proposed formulas yield
only approximate danping characteristics .
Danping tests can be carried out by one of two methods:
1) determination of the dependence of the force of hydraulic resistance
on the rate of travel of the rod;
2) determination of the energy absorbed by the shock absorber in the
presence of harmonic vibrations of the rod.
When conducting tests by either of these methods, the air (or nitrogen) /^
should be drained from the shock strut, since
only the danping forces are to be determined in
the test.
The test procedure by the first method conr-
sists in measuring the steady rate of travel of
the rod of the shock absorber under the effect of
a constant load at various values.
In the second method of testing, harmonic
vibrations are inparted to the rod of the shock
absorber on a special rig with a rotating eccerv-
trie. The variable axial force in the shock ab-
sorber is measured at different values of the
vibration anpHtude and frequency (revolutions of
the eccentric) of the rod.
For a landing gear with vertical struts
Fig.3^26 Vibration Anpli-
tude Dependence of Equiva-
lent Danping for Shock
Absorber with Dry P^iction
and Quadratic Ifydraulic
Resistance.
317
( Fig .3. 17, "b) J direct tests of the tire-oleo system should tie carried out. It is
desirable to make such tests also for a pyramidal lariding gear (see Pig.3#17,a).
Here, the shock ahsorter connected in sequence "with the tire can "be tested in
the same manner as that used for a landing gear -with vertical struts except that
a special tire identical -with that used on the helicopter is selected, whose
stiffness is greater ty a factor of n than that of the tire corresponding to a
pyramidal landing gear. The value of n is calculated "by means of the formula
-m
where Ps.a is the force in the shock absorber under the vertical force P^^ on
the tire.
2. Effect of Locking of the Shock Absorher as a Consequence
of I^ictional Resistance of the G-land and Self-Excited
Vibrations of the Helicopter
The force of friction Fq in the packing (glands) of the shock absorber for
all practical purposes is independent of the rate of motion of the rod [eq.(3.l)].
Therefore, the effect of friction in the gland is analogous to the effect of so-
called dry (or Coulomb) friction.
This leads to the effect that, in the presence of small vibrations at a
variable force P < Pq, the shock absorber does not operate and behaves like a
rigid rod. Therefore, at a sufficiently small vibration anplitude of the hell-
copter the shock absorbers are inoperative and only the tires, which are virtual-
ly deprived of danping, act as elastic components of the landing gear system.
If the angular velocity of rotor rotation lies within the instability zone
of the helicopter with inoperative shock absorbers, the position of equilibriimi
of the helicopter, generally speaking, will always be unstable and small heli-
copter vibrations of increasing airplltude are sure to arise. Upon an increase
in airplltude of the vibrations, the variable force in the shock absorber also
increases. At a certain vibration anplitude a"'^", the force P in the shock ab-
sorber becomes equal to Pq . At large vibration ajiplitudes of a > a"^" the force
P > Pq, and (if T < T^j.) the shock absorbers begin to operate.
If the shock absorption danping is properly selected, self-excited vibra-
tions with a certain constant small anplitude a, greater than a"''", are generated
in the system.
Thus, for any helicopter within the unstable range with the struts in- /289
operative, there mil always be self-excited vibrations caused by the effect of
dry friction in the landing gear shock absorbers.
Such self-excited vibrations should never be confused with ground resonance
in the conventional meaning of the term. Self-excited vibrations are safe and
may arise even when the margin for ground resonance (at large displacements) is
sufficiently great.
318
In the most common designs of shock absorbers (oleo-pneiomatic struts), the
friction in the gland is relatively great so that, in calculating the airplltude
of such self-excited vibrations, only the danping caused by this fraction need
be allowed for and the forces of
hydraulic resistance in the shock
^^ ^ a ^ Gl ^.,^„fL.H n absorber can be neglected. With
such an approximate calculation,
the anpHtude of self-excited vi-
brations can only be greater than
>5 2 J _^ — J,/? the actual airplitude#
Fig #3. 27 Diagram of In-Series Connec-
tion of Tire and Shock Absorber -with
Dry Friction.
1 and 2 - Springs; 3 - Piston.
To estimate the ajipHtude of
self-excited vibrations, we can de-
rive certain sinple formulas.
Let us examine a system con-
sisting of two springs (l) and (2)
connected in series, one of which
with a stiffness Cp^ simulates the tire and the other with a stiffness Cs.a> the
shock absorber (Fig .3 •27)* Some element [piston (3)3 with dry friction char-
acterized by the force Pq is connected in parallel with the spring (2).
Under the effect of a force P(t) varying in time according to a certain
law, let the system execute vibrations such that the point A whose displacement
we denote by s will execute the harmonic vibrations
s==^SocospL
(3.7)
If the anplitude Sq is small, the spring (2) does not work and the spring
(1) will s-uffer a deformation Si = s varying in accordance with the harmonic
law (3*7) • In this case, the force P also varies in obedience to the harmonic
law
However, the work done by the system will be of this type only if Pj^^^x "^ ^0 ^^
thus if So <
As soon as Sq >
Cpn ^vn
-, the spring (2) starts moving. Here,
there are certain time intervals when the spring (2) operates [sUding in the
element (3)] and time intervals when the spring does not operate.
Let 6 = s - Si be the deformation of the spring (2), which we will consider
as positive if the spring (2) is compressed. Then, the dependence of the comr-
pressing force P on the quantity 6 can be written in the form
/> =
'^^^S + P,, At
8>0;
A.»S-/>o, At 6<0.
(3.8)
In the presence of oscillatory motion, the dependence P = P(S) has the form
of a hysteresis loop (Fig .3. 28).
319
When the quantity 6 reaches a maxini-um value of a and remains constant /29Q
after this (6 = a), force P can take any value in the interval Cs,aa-P^P<
^ Cs.a^ + Pq.
The relation 6(t) for time intervals corresponding to sUding (6 ^ o) can
be determined from the equation
c^iso^os pi ~K)==c, J ±P^,
which expresses the equality of forces on iDoth elements (l) and (2).
Fig. 3 •28 Hysteresis Loop for
Shock AlbsorlDer with Dry
Friction.
Fig '3 '29 law of Time Rate of Change
of Forces and Displacements in
Shock Absorlber.
From this equation, we find 6(t) for the sliding sections:
Pa
S (0 = -— cos pi± -—
For sections where sUding is alDsent, we have 6 = ±a.
Figure 3*29 gives graphs of the time rate of change of the quantities s(t).
6(t), and P(t) for the case Cg.a = ^In ^^ Sq = 2
The magnitude of the
'pn
vibration anpHtude a of the shock absorber rod can be found from the e^^ression
for 6(t), if we set there cos pt = 1. This will yield
(3.9)
The work done by the frictional force during vibrations can be determined
from the formula
320
^Po'
-APaa.
Let us conpare the system shown in Eig.3»27 "with some equivalent linear
shock absorber which, at an anplitude of the rod Sq, absorbs the same work and
has the same value of maxlmxm force P^ ax •
Equating the e^spression for work done by the linear shock absorber (A =
= rrk^qpsl) to the work done by the fractional force Ap^ , we obtain the follow-
ing expression for the danping coefficient k^^ of an equivalent linear shock /291
absorber:
A».=-
i-W
f « P(c^,.+ «s.a)
(3.10)
where Tj is some dimensionless coefficient depending on the vibration anplitude sq
and determined tjy the formula
Sq 52
— C^
^0
(3-11)
(3.12)
The expression for stiffness c^q of an equivalent linear shock absorber is
obtained on comparing the value of the maximum force for the linear (Pnax =
= CeqSo) and nonlinear [P^^x = cJn(so - ^)^
shock absorbers:
0,2
0.1
^mo
■
r
_ 1
—
,
-^
/
i-
1
S^^^n
Po
-So
• + 1
(3.13)
Fig .3 .30 Dimensionless
Datrping Coefficient T| as a
Function of the Relative
Vibration Anplitude Sq •
Figure 3*30 shows the dependence of the
quantity T] on the dimensionless vibration ampli-
tude So • The quantity T| reaches a maximum
value T\n(
= i at
= 2. At So > 2, the danp-
ing drops with an increase in vibration arrrpli-
tude.
The maximum value of k-
k^q"" is equal to
^'.r,r
(3.1!f)
321
A conparison of this value with the value of k^q^ olDtained for the linear
tire-oleo system [see eq.(2.38)] shows that the shock alDsorber with dry friction
in a system with a pneumatic tire produces, lander the same conditions, a maxi-
mum damping lower "by a factor of tt/2 than the linear shock absorber.
Thus, the atiplitude of self-excited helicopter vibrations caused by friction
in the packing of the shock absorbers can be found from the condition (3. 10):
yfe-? ^k^^ ==~
{<^U
Pi<
r\.
pn
where k^®"^ is the danping required for elimination of ground resonance •
Prom this equation, we determined the corresponding value of T| and then,
from the graph in Fig, 3 .30, the corresponding value of Sq •
3* Characteristics of Blade Dampers and their Analysis /292
In our presentation of methods for calculating ground resonance, we pro-
posed that the drag hinge danpers have linear characteristics, i.e., that the
moment of the daiiper M is proportional to the^angular velocity | of rotation of
the blade relative to the drag hinge: M = k^l.
M
M
%
a)
i)
M
M^
ir^
c)
Fig .3 .31 Typical Characteristics of Blade Dairpers-
a - linear danpers; b - Friction danpers;
c - Stepped danper.
Actually, the characteristics of blade danpers are nonlinear as a rule.
Two types of danpers are predominantly used:
1) hydrauHc danpers;
2) friction danpers.
Ifydrauli-c danpers may have different characteristics depending upon design.
In particular, a hydraulic danper can be linear (so-called laminar danper, char-
acteristic a in Pig .3 •31)' However, Unear danpers are used extremely rarely,
since they have serious shortcomings.
One of the shortcomings is the great sensitivity of linear danpers to tem-
322
peratiire, which is explained "by the fact that dairping in such danpers is pro-
portional to the viscosity of the hydraulic fluid which is greatly dependent
upon tenperature»
Another shortcoming of linear danpers is that the moment of such a darrper
is proportional to the "blade vibration frequency. Actually, if a blade executes
harmonic vibrations relative to the drag hinge 5 = §o sin vt, ,then the moment
of the linear danper varies in accordance with the law M = k^? = vk^lo cos vt.
This causes the linear danpers, during forward flight of a helicopter, to
load the root portion of the blade with large bending moments, since the blade
vibration frequency in flight is by a factor of about 4 greater than at ground
resonance.
This drawback is largely absent in the most widely used hydraulic dampers
with a stepped characteristic (see Fig.3.31, c) and also in friction danpers (see
Fig.3»31,b). The point A on the characteristic curve of the stepped danper cor-
responds to the instant of opening of special valves.
The characteristic of a friction danper (see Fig.3.31,b) can be regarded as
a particular case of a stepped characteristic.
To calculate ground resonance of a helicopter with nonlinear blade danpers,
the latter can be replaced by some "equivalent" linear danpers whose coefficient
of danping depends upon the anplitude and frequency of blade vibrations. The
coefficient k^q of such an equivalent linear danper can be determined from the
condition of absorption by this danper of the same energy per oscillatory period,
at a given harmonic vibration anplitude and frequency, as is absorbed under /293
the same conditions by a nonlinear danper. For a friction danper, we have
'^ 71 Vco
where
Mo = tightening moment of the danper (see Fig .3 .31, b);
5o - anplitude of blade vibrations;
V = frequency of blade vibrations.
From the same formula, we can determine approximately the value of k^^ for
hydraulic danpers with a stepped characteristic, if the latter is close to the
characteristic of the friction danper.
In the general case, the quantity k^q can be determined from the known
characteristic M(5) of a nonlinear danper by means of the formula
T
"^^o i (3.16)
at
i =>^Q sin vi^ndT = ~^
323
For a manufactured dauper, the quantity kg^ can "be deterrained also experi-
mentally in special laboratory tests. In such tests, harmonic vibrations are
inparted to the rod of the dairper and the magnitudes of the damper moment are
recorded on an oscillogram.
The main shortcoming of danpers with a stepped characteristic and, in par-
ticular, of friction danpers is the presence of a so-called excitation threshold
for helicopters equipped with danpers of this type,
A helicopter which is stable at small vibration
anplitudes may become unstable at large vibration
aiiplitudes exceeding the excitation threshold.
Let us exajnine this phenomenon for the exairple
of a friction dairper. Figure 3*32 shows the de-
pendence of the work A absorbed during one oscil-
latory period by friction danpers (curve a) and
linear (curve b) danpers on the blade vibration am-
plitude 5o (^"t constant vibration frequency). For
the friction danper, the graph A (go) is represented
by a straight line, whereas for the linear danper
it forms a parabola. As result of calculating the
ground resonance, let the value k^®*^ of the required
daiiping of the blade be determined for the case of
a linear danper; then, the coorve a in FLg#3#32 cor-
responds to this value of k^, whereas the ciorve b
corresponds to the available danping of the friction
danper actually lashed-up to the helicopter.
Fig .3 '32 Work Absorbed
per Oscillatory Period
as a Function of Aipli-
tude, for a Danper with
Dry Friction and a
Danper with Linear Char-
acteristic.
^Let these curves intersect at a certain point c,
corresponding to the anplitude gf . Then, during blade vibrations with an anpli-
tude ^o ^ Vo9 "tihe danping provided by the friction danper will be greater than
required, whereas during blade vibrations with^ an anplitude §0 > ?'S> the danping
will be inadequate. The vibration anplitude ^f also represents the excitation
threshold. The value of 53 can be determined from eq.(3.15)«
Thus, if the helicopter suffers some perturbation (shock) which sets up f^^h
vibrations (both of the helicopter and of the blade) then, if the blade vibra-
tion anplitude is less than 53, the motion will be stable and the vibrations
will die out. If the perturbation is sufficiently great (§0 > §o):f then increas-
ing helicopter vibrations will occur.
The presence of an excitation threshold for helicopters with blade danpers
of stepped characteristics is a serious shortcoming.
There are quite a few cases known where a helicopter that has been in
service for a long time underwent ground resonance as the result of some severe
shock, usually as a result of a rough landing with only one wheel of the main
landing gear making contact with the ground.
324
This main shortcoming of nonlinear dairpers can Ids conpletely eliminated only
"by using danpers that pro-vide considerable dancing at low "blade vibration fre-
quencies (ground resonance) and slight danping at a vibration frequency equal to
the rotor rpm (and higher),. In particular,
^ ^ such a danper can be Unear. Figure 3-33 shows
the diagram of a linear danper of this type.
The dairper consists of an elastic element of
stiffness c and of the danper proper with a co-
efficient k, connected in series.
/
^J-M/WV-<^-
Fig. 3 .33 Diagram of Element
in which the Elastic Element
and Danper are Connected in
Series .
The characteristics k and c of this danper
can be selected such that, after ensuring ade-
quate darrping at ground resonance, there are
small bending moments on the blade in forward
flight of the helicopter (see Sect .6). For the
calculation, such an element can be replaced by some equivalent element of stiff-
ness c^q ^^^ ^ dairping coefficient k^q determined by means of the formulas
S='^
k.^^k
1 +
if)'
(3.17)
These foxTiiulas are obtained in the same manner as eqs«(2.33) and (2.34).
4. Effect of Fla-piping Motion of Rotor on Ground Resonance
As ala:*eady stated, the coninonly used blade danpers are nonHnear. The main
feature of any nonlinear danper is that, if the motion of the blade consists of
two harmonic coirponents, the danping of one of these conponents will depend on
the anplitude and frequency of the other harmonic conponent, whereas a linear
damper absorbs the energy of each of the harmonic coirponents regardless of the
magnitude of the other.
This featiore of nonlinear danpers e^cplains the following important phenome-
non which has long been noted in helicopter tests: When a helicopter is oper-
ating on the ground, ground resonance may be caused by smooth deflection of the
cyclic pitch control stick from the neutral position. If the stick is then re-
turned rapidly to the neutral position, the vibrations die out. This phenomenon
is utilized in e^q^erimental tests of helicopters for ground resonance. The /295
phenomenon is analogous to the effect of flapping on the occurrence of flutter.
Let us examine the mechanism of this phenomenon for the case of a friction danper
(see Fig.3.31,b).
Let us first attack the following abstract problem: let some body A slide
with a velocity V along some plate B (see Fig .3 .34) which is executing harmonic
vibrations in a horizontal direction according to the law y = yo sin cot.
325
The ^DOdy A is forced against the vi"brating plate B tjy a certain normal
force N. We will assxame that the friction "between the surface of the body A and
the plate B corresponds to the ideal law of dry friction, i.e., the force of
friction is constant in magnitude and equal to Pq = |jlN where p. is the coefficient
of sliding friction.
?777\ *-
^/A
7777X
y=yo slri wt
p?=^
p^
\h
T '^r
r,.
•*■ ■ 1 i if ■-
1 1
^27-,-
t
T
Fig -3 -34 Diagram of Body Motion
along a Plate Vibrating in a
Horizontal Direction.
Fig .3 .35 I^w of Time Rate of Change
in Relative Velocity and Force of
Friction during Uniform Motion of
the Body along a Vibrating Plate.
The direction of the friction force depends upon the direction of the rela-
tive velocity of the body A in conparison with the plate B.
Let us assiime the friction force P applied to the plate as positive when
directed opposite to the absolute velocity of the body A, i.e., to the right.
The displacement y of the plate B will be considered as positive when directed
to the left. Then the law of friction can be written as
_| + P, at V>y;
'Po at V<y.
For the relative velocity of motion of the plate, we have the expression
V^^j -^V~~y = V-~ o)r,'o COS iyyf.
Figure 3*35 gives the graph of this dependence. The relative velocity as a
function of time is depicted by a cosine curve shifted by an amount V along the
ordinate.
The graph indicates that, at V < uoy©, during one oscillatory period T the
time interval (T - 2Ti) during which V^.^^ is positive is greater than the time
interval 2Tx during which Vj.e i is negative. Below this graph, we show a cor-
responding graph of the time dependence of the friction force P. During one os-
cillatory period, the friction force for a certain time interval 2Ti is directed
326
to the left (negative), while dtiring the time interval (T - 2Tx) it is directed
to the right, opposite to the direction of motion.
Thus, dioring motion of the body A (see Fig ,3 •34) along a vibrating plate B,
the friction force periodically varies its direction only if V < ouyQ . In. this
case, the friction force for the greater part of time is directed opposite to
the motion so that, on the average, the friction offers resistance to the motion
of the body A.
For a uniform motion of the body A to the left, we must apply to it a /296
time-variant force which, at each instant of time, would balance the friction
force* Let us calculate the average value Pay of this force during the period,
understanding by this a constant force which, during one oscillatory period, does
the same work in absolute displacement of the body A as the actual force of fric-
tion. If the mass of the body A is infinitely great and if vibrations of the
body A can be neglected and if, in addition, its motion on the vibrating plate
can be assumed as loniform, then the force Pay will represent the actual force
needed for a uniform motion 'of the body A.
The work done by the average force per oscillatory period of the plate will
then be equal to
The work done by the friction force during one period will be equal to
Afr==PoW-2T,]-PoV[2T,]^PoV[T~4T,l
Equating these two values of work Af^. and Aav* we obtain the following ex-
pression for the average motive force:
^av — "o
[>-^l
T
To determine the value of — ~ we note that
T
cos(ori=-^ .
2tt
Hence, taking into account that T = , we obtain
0)
II
T
.^yol'
Consequently,
L n K'^yoJ.
327
The obtained e^qDression holds if and only if V < cuyQ. At V > cwyo^ the fric-
tion force -will not change either magnitude or sign, renaining equal to Pq. Tak-
ing into account the aforesaid and introducing the dimensionless average force
F^^ = ■ l^^r ^ we obtain the following expression for the latter:
Pq
1 -A ,,,./ (v:
1, if V>^yo
Q' '*/ ^<^^o;
(3-18)
Figure 3-36 gives the graphs of the dependence of P^v on the dimensionless
velocity V = .
At a low relative velocity V of the plate, we can use a sirrplified Unear
dependence_Pav('V) which is obtained if, in the expansion of cos"" V in a power
series in V, we Hmit ourselves to the first two terms. In this case, we have
r-^y ' — - .
Jt tl)£/o
Thus, during slow motion of a body over a rapidly vibrating plate, the /297
average force of friction can be considered approximately proportional to the
first power of the velocity
Pa,^=ke^y, (3.19)
where the proportionality factor is
S=T-;S- (3.20)
The above statements indicate that, under the examined conditions, dry fric-
tion is in a sense equivalent to Unear viscous friction, the equivalent damping
coefficient being inversely proportional to the vibration frequency and anpHitude
of the plate. This important fact was first noted by Heinrich (Ref .41) and
checked experimentally by A.A.Krasovskiy (Ref.l4).
It is obvious that, during slew harmonic vibrations of the body A over a
rapidly vibrating plate B, it is also possible to approximately calculate the
danping of these vibrations, making use of eqs.(3*19; and (3.20).
Thus, if in an element with dry friction the relative motion of the rubbing
surfaces represents the sum of two harmonic vibrations, one low-frequency and
the other high-frequency, then the danping of the low-frequency vibrations can
be calculated approximately by using eqs.C3*19) and (3 -20), understanding by cu
and Jo the frequency and anplitude, respectively, of the other (high-frequency)
harmonic coirponent.
328
let us now return to an examination of "blade vit>rations of a helicopter
equipped with a friction danper lashed to the drag hinge* Upon deflection of the
pitch stick during operation of the helicopter on the ground, blade flapping
relative to the flapping hinges takes place • As is knovm, the blade flapping
angle P will then vary in time according to the hai*monic law:
p = aQ — ajcos (0^ — ^jsinoj^,
where
sl-i and bi
coning angle;
flapping coefficients.
'■° ^=4
Dioring blade flapping, Coriolis forces arise
which cause blade vibrations relative to the drag
hinge. The anplitude of the first harmonic 5 i of
blade vibration relative to the drag hinge can be
determined from the well-known formula (Ref .^8)
^1 —
^0
(3*21)
Fig. 3 ,36 Relative Aver-
age Friction Force as a
Function of the Dimension-
less Velocity of the Body
over a Vibrating Plate.
the blade at ground resonance
formula
where
As already explained in Sections 1 and 3,
dioring ground resonance (in the case p^.^ = O) the
blades vibrate with a frequency Vq^u (vq ^ 0.25),
i.e., with a frequency about four times lower than
the frequency of forced blade vibrations caused /2^8
by flapping. Therefore, in conformity with the
foregoing statements, the dajrping moment acting on
can be calculated approximately by means of the
M = k,,'z,
(3.22)
(3-23)
The quantity Mq represents the tightening moment of the friction dairper
(see Fig.3.31,b).
Thus, forced vibrations in the plane of rotation of a blade with a friction
danper, caused by flapping of the blade in the thrust plane, lead to an effect
equivalent to the introduction of a linear dairper in the drag hinge whose danp-
ing coefficient [eq.(3-23)] is inversely proportional to the anplltude 5i of the
forced blade vibrations relative to the drag hinge. Therefore, all statements
on the excitation threshold of a helicopter with friction dairpers (Subsect.3)
hold only when there is no blade flapping. This usually occurs when the rotor is
operating at low rpm. Consequently, the excitation threshold must be estimated
for ground resonance in terms of the first oveartone. Equation (3*23) should be
used for estimating groxond resonance in the presence of rotor flapping (at the
operating rotor rpm) . This is especially iiiportant when calculating ground reso-
329
nance of a helicopter during the ground run, which vdll be taken ip in Section 4*
Section 4« Ground Resonance of a Helicopter during Ground Run
In presenting methods of calculation for natural lateral vibrations of a
helicopter on the ground (Sect. 2), it was assumed that there was no wobbling of
the pneumatic tire on the ground surface.
When a tire wobbles its lateral stiffness diminishes, while the vertical
stiffness remains unchanged. A decrease in lateral stiffness of the tire during
such rocking and a certain additional dajotping during lateral displacements of
the wobbling tire can be determined on the basis of the existing theory of shimn^r
of castoring wheels* A method of such a calculation is given in this Section.
Any decrease in lateral stiffness of the tire on wobbling (decrease in the
quantity c^^^, see Fig. 3.1?) leads to a decrease in the natural vibration fre-
quencies of the first and second overtones and thus to a reduction of the bound-
aries of the corresponding unstable range. As indicated above (Sect .2, Sut^
sect. 7) 5 for a single- rotor helicopter the instability zone, corresponding to
the second vibration overtone, is above the operating rpm of the rotor, and the
margin with respect to rotor rotations is sometimes no more than 30^. A reduc-
tion of the unstable range during tire wobbling may be of the order of 20 - 30^.
Therefore, it might happen that a helicopter, which is stable when operating in
situ, becomes unstable during the ground run. In this case, we can speak of a
critical ground speed of the helicopter at which motion becomes unstable.
1. Stiffness and DamDing;_ of a Wobbling Tire
let us examine a tire uniformly rolling along the ground ( Fig .3 '37) • Let
the wheel execute lateral vibrations in obedience to a harmonic law, such that
the axis of rotation of the wheel remains at all times parallel to its initial
position and the distance from the axis to the ground remains constant.
We will select a stationary rectangular coordinate system zOs lying on /299
the ground surface, with the Oz-axis being parallel to the wheel axis. Let the
lateral displacement z of the diametral plane of the wheel vaiy in time in ac-
cordance with the harmonic law^'"":
z=.Zoe'^', (4.1)
where
Zq = vibration anplitude;
uo = angular vibration frequency.
Let us then determine the lateral force P^ exerted on the tire by the ground
during its motion.
■"" We have in mind that the actual displacement is the real part of the indi-
cated conplex expression. The use of conplex e^q^ressions in deriving basic for-
mulas permits an appreciable sinplification of the calculations.
330
Let X "be the lateral deformation of the tire, i.e., the distance between
the diametral wheel plane and the point of the tire which forms the center of
the contact area before lateral deformation.
equal to
Then, the lateral force P^ will be
2 yDiametral plane
of tire
Wobble
line
cf^l,
(4.2)
where c^^ is the lateral stiffness of the tire
in the absence of wobbUng.
Furthermore, let s be the path of the
tire reckoned from the Une of wobbling and cp
the angular deformation of the tire, i.e.,
the angle between the line of intersection of
the diametral wheel plane with the ground
surface and the tangent to the material line
belonging to the tire surface and representing
the Une of intersection of the diametral
wheel plane with the undeformed tire surface.
The quantities z, X, and cp are related by
the so-called conditions of wobbling which, in
conformity with M.V.Keldysh's hypothesis
(Ref .15), have the form
Fig .3 .37 Planview of Contact
Surface and line of Tire
WobbUng.
cp - Angular deformation of
tire; \ - Lateral deformation
of tire ("tilt mo-
tional methods.
dz ,_ dl
ds "^ ' ds
ds
(4.3)
Here, cy and P are certain constants for
a given tire, which can be found by conven-
Changing to derivatives with respect to time t and taking into account that
s = Vt (where V is the velocity of the tire in the direction of the Os-axis),
we obtain
at
(4.4)
Putting
and
X^X.^'^^
9=^90^'*"'
(4.5)
331
and taking into account eq.(4»l), we obtain from eqs.(4-4) the follovdng ex- /gOO
pressions for determining the constants Xq and cpo*
Hence,
Substituting the foiind value of Xq into the first equation of the system
(4*5) and then into eq.(4»2), we obtain the following expression for the force P^
exerted on the tire by the ground:
where
We will designate the conplex quantity
zo ^ . , aV^ (4.8)
the lateral conplex dynamic tire stiffness in the presence of lateral harmonic
vibrations of the wheel.
The modulus of the conplex dynamic stiffness represents the ratio of the
airplitude Pq of the lateral force to the vibration anpHtude Zq of the diametral
plane of the wheel. The argument of the conplex quantity D(a)) represents the
vibration phase of the force P^ with respect to vibrations z of the wheel.
Furthermore, let us examine some Hnear elastic element with danping (see
Fig .3. 22). The force P acting on this element and its deformation s (stroke of
the element) are connected by the relation [eq.(2»25)3
Let los introduce the concept of conplex dynamic stiffness of such an ele-
ment and establish its connectivity with the coefficients c and k of its stiff-
ness and danping.
Let the displacement s of the elastic element vary in time according to the
332
i
harmonic law s = Soe^^^ . Then the force acting on this element -will vary also
"by the harmonic law:
or
where
We designate the quantity
PQ^ip + i^^k)sQ.
D(oy)^c-\rti^k (/^^9)
the coirrolex dynamic stiffness of . an elastic element with dairping. As shown ^DJ
eq»{k*9), the real part of the conplex quantity 0(00) represents the stiffness co-
efficient c of a spring, while the imaginary part represents the danping coef- /301
ficient k of the element multiplied by o).
To calculate the natural vibrations of a helicopter on the ground in con-
formity with the scheme depicted in Fig .3. 16, the characteristics of elasticity
and danping of the elements c^ and Cy must be properly chosen.
It is obvious that, to calculate vibrations of a helicopter during ground
r-un, it is sufficient to select the horizontal elastic elements (c^^ in Fig .3 •I?)
such that their conplex dynamic stiffness will be equal to the conplex lateral
stiffness of the tire in wobbling. The coefficients of stiffness and danping of
the thus selected "equivalent" elastic element can be determined, respectively,
as the real and imaginary parts of the conplex quantity D(ud) which is expressed
by eq.(4.B).
Separating the real and imaginary parts in eq.(4»B), we obtain
^e? "^2 / aV^Y p2^' (4-10)
\ a>2 y 0)2
r.P"
W \ (aV^ \
m
k fL, — Kj^_i1j^1__ (4.11)
'} CO / aV2 \2 |2y2
(
6)2 J Ci>2
The resultant formulas are conparatively coirplex and require knowledge of
the tire constants 01 and P •
The formulas for determining c^d and k^^ can be greatly sinplified if we
replace M.V^Keldysh's conditions of wobbling by the so-called "tilt" hypothesis,
according to which the lateral deformation of the tire X ("tilt") is connected
333
1.0
0.5
i
(ti'lB ijsec
^^
'Py'^SOOkg
3QQkg
Acca* to
tilt
theory
\
N
\
«**^
^^
^
iQ
10
30 Vkm/hr
•with the angular deformation of the
tire cp by the siirple relation
^-==^^' (4.12)
where Tj = -H^ is the so-called
a
"tilt" coefficient.
As shown in a paper "by M,V.
Keldysh (Ref.l5), this coefficient
is approximately equal to the radi-
us r of the undeformed tire
a
(4-13)
0,2
0.1
300kg
1
Wks^
Accd, to tilt
iSSSBu
----^
/
.w,^
10
20
30 Vkm/hr
Fig. 3 .38 Relative Lateral Stiffness and
Lateral Dairping of lire as a Function of
Helicopter Ground Speed ('o) = 18 \ .
\ sec /
(r being the distance between the
tire axis and the ground in the ab-
sence of conpression) .
In this hypothesis, the first
condition of the system (4 .4)
yields
dt~ y\ ~^ dt
Putting, as before,
in
0.6
DA
0.1
X
" • 1
i
\i
1
>
s>^
/
V
""^
- ■
^r=-
/
"^~■
\//pr^\/
Fig.3.39 Relative lateral Stiffness
and Lateral Danping of Tire as a Func-
tion of Dimensionless Helicopter Ground
Speed.
= ^n^^'
= An^''
V"
we obtain
D(a)) = ^ = cf
^0
1 +
(4.1!f)
On separating the real and /303
imaginary parts in this expression,
we obtain the following eijq^ressions
for the stiffness and danping co-
efficients of an equivalent elastic
element :
'"-' w (4.15)
Vt]o) ;
334
nP"
1 +
\T)a> /
The siirplified formulas (4»15) and (4»16) are more convenient for practical
calculations and do not require knowledge of the tire constants of and. p. The
accuracj of the approximate formulas is fully sufficient for practical purposes.
This is demonstrated hy the comparison graphs in Fig.3*3^j which were obtained
from calculations of the main landing gear wheels of the Mi-1 helacopter (uo =
= 18 — = — corresponds to the frequency of the second overtone of vihrations),
sec
using Keldysh's theory and the tilt theory.
Thus, in calculations of natural helicopter vibrations during ground run it
is expedient to use eqs.(4»15) and (4»16). In this case, the quantity od in
these equations must be substituted by the frequency p of the lateral helicopter
vibrations. Figure 3 •39 presents graphs of the dependence of the dimensionless
lateral stiffness f — ^^— ) = c^^^ and dajrping f — ^ — j on the dijnensionless rela-
tive velocity V = = . As we see from the graph, the lateral stiffness
PTI pr
of the tire largely depends on the helicopter speed. At V = 3 (which, for the
tire of the Mi-1 helicopter at p = 18 is about 50 km/hr), the lateral tire
sec
stiffness is by a factor of 10 less than that of a stationary tire.
2. Calculation of Ground Resonance and Results
The calculation of ground resonance during ground run can be performed in
the conventional manner (Sect. 2), except that the values of lateral tire stiff-
ness c^^, when calculating the natural vibrations, should be replaced by the
values of o^^ derived from eq.(4*15); when determining the coefficients of dairp-
ing of the natural vibrations (Sect. 2, Subsect.4), the additional danping of the
horizontal elastic elements (see Fig .3.1?) should be taken into account in con-
formity w?.th eq.(4*l6). In this case, the values of uo in eqs.(4*15) and (4*16)
must be substituted by the values of the frequency p of the corresponding vibra-
tion overtone of the helicopter. Such a calculation method is couplet ely justi-
fied since, at the boundaries of the instability zones, there are purely har-
monic (londairped) vibrations, and eqs.(4»15) and (4»16) are derived precisely for
the case of harmonic lateral tire vibrations. The purpose of calculating ground
resonance is to find these boundaries of the unstable range.
When using the formula
335
o/""-
1 +
[^1
(4.17)
in the calculation of natural helicopter vilDration frequencies, a difficulty is
encountered connected with the fact that, to find the natural vibration fre- /^04
qaency p the value of 0^^, must "be known which, in turn, depends ipon p. There-
fore, the calculation of* natural vibrations (determination of p) should be car-
ried out by assigning different values to c^jq in the interval < Cg^ < c^^ ,
and then, after determining p, deriving the corresponding value of the ground
run speed from eq.(4*l7)» Here, for determining this speed we have the formula
V
='"/;
cr
(4.18)
Based on such a calculation, it is possible to construct the graph of the
dependence of the instability zone boundaries on the ground run speed V. Here,
Pig ,3 .40 gives the results of a calculation of
ricrf^'^p^ this type for the Mi-1 helicopter. The graph
shows the lower boundary of the instability zone
corresponding to the second vibration overtone
as a fxinction of the ground run speed V#
^00
300
zoo
^
■\
—
=^
^-v_
\,
'^cr
^^///}
1
yZo Operating
rpm t
10 20 30 ^0 Vkmjhr
Fig .3 #40 Lower Instability
Zone Boundary as a Function
of Ground Run, for the Mi-1
Helicopter ♦
n^r - Critical revolutions
corresponding to onset of
self-excited vibrations;
n""" - Revolutions correspond-
ing to the instability zone
center.
As indicated by the graph, the critical
revolutions n^^j. of the rotor corresponding to
the onset of ground resonance appreciably de-
crease with an increase in helicopter speed* If
for a stationary parked helicopter the rpm margin
is 36^, this margin will decrease to &fo at a
ground run speed of 60 km/hr.
It is iirportant to note that, ipon an in-
crease in speed 7, the graph of n^^ approaches
some asymptote. This has the following physical
meaning :
Upon an increase in V, the lateral stiff-
ness of the tire determined 'bj the quantity C^q
[eq.(4»17)] decreases without bounds, approach-
ing zero. In this case, the natural vibration
frequencies of the first and second overtones
decrease, with the frequency p of the first
overtone approaching zero and the frequency of
the second overtone tending to the value
(4.19)
336
The quantity po represents the natural lateral vibration frequency of the
helicopter in the absence of lateral tire stiffness.
The vibration mode of the helicopter corresponding to this frequency repre-
sents the rotation of the helicopter "body about the principal longitudinal axis
of inertia. The corresponding vibration node (see Sect. 2, Subsect.3) coincides
with the center of gravity of the helicopter.
A situation of this type might occur for a helicopter standing still or
moving over a smooth surface of ice, where it can be assumed that there is no
friction between tire and ground (here, also c^^ = 0)«
This results in the possibility of a siii?)lified (estimate) calculation of
ground resonance diiring the ground run, when the natural vibration frequency /305
is determined from eq.C4«l9)« Here, for the mass of the equivalent elastic base
(Sect #2, Subsect.6) we have the formula
'"«^=ir- (4.20)
For the danping coefficient of the helicopter we have
n^^^flJ^+^, (4.21)
where H is the distance from the ground surface to the center of gravity of the
helicopter.
The quantity kg^ is determined by means of the formula
c
pn
\rpQj
[rpj
The quantity ky is the danping coefficient of the vertical elastic elements
(see Pig .3. 16), which depends on the dairping properties of the shock struts of
the landing gear and is determined in the same manner as that used in Section 2,
Subsection 5*
Such an approximate calculation for a helicopter having an unstable range
located above the operating rpm produces a small error "in the safety factor".
3. Ground Resonance on Breaking Contact of the Tires
with the Ground
All above methods of calculating ground resonance presumed linearity of the
tire characteristics. However, in reality the tire characteristic (even approx-
337
imately) can "be considered linear only in situations in which the tire, dull-
ing its deformation, remains in contact mth the ground surface. In general,
the characteristic curve of the tire has a slope as shown in Fig .3 .41*
If Py is the force exerted by the ground on
the tire and Sy is the corresponding displacement,
the characteristic of the tire has the form
Py =
CuS.
v'^'y
at
at
Sy<0.
If the investigation covers only small vibra-
tions of a helicopter near a position of equilih-
rixmi corresponding to the given rotor thrust T at
which Pv = Po
Fig .3 .41 Nonlinear De-
pendence of the Force
Exerted by the Ground on
the Tire on the Vertical
Displacement of the Wheel
Axis.
and Sy =
So, so that the point of
the state of the tire dioring vibrations comes to
lie on a certain segment AB wholly within the
linear portion of the characteristic, then all cal-
culation methods based on linearity of the tire
characteristic are valid (for such small vibra-
tions) .
However, at large vibration anplitudes it may
happen that the point on the diagram depicting the state of the tire is beyond
the limits of linearity of the characteristic. Obviously, this will be the case
whenever the anplitude of displacement As is greater than the airplitude of static
conpression Sq* The extent of static conpression Sq, just as the force Pq, /306
depends on the rotor thrust and decreases with increasing rotor thrust T, ap-
proaching the magnitude of the helicopter weight G-. If T < G-, the tire is forced
against the ground; however, the helicopter vibration anplitude at which the
tires begin to break contact with the ground is smaller the closer the quantity T
approaches the value T = G. Therefore, breaking contact of the tires on takeoff
and landing is most readily achieved when the rotor thrust is less than the
helicopter weight but still si:ifficiently high.
Calculation of helicopter vibrations on breaking contact with the ground is
rather coirplicated. However, without actually performing such calculations,
certain valuable but qualitative conclusions can be drawn. Actually, during vi-
brations on breaking contact of the tires, the helicopter represents a nonlinear
oscillatory system with backlash. It is known that the natural vibration fre-
quency of a system with backlash depends on the vibration atrplitude and on the
magnitude of backlash; the greater the backlash (at a given anpHtude), the
smaller the frequency of natiiral vibrations. This is physically understandable,
since the presence of backlash is equivalent to a decrease in the average (during
one oscillatory period) stiffness of the elastic element.
Consequently, during helicopter vibrations on breaking contact with the
ground the natiiral vibration frequencies decrease, acconpanied by a reduction in
the extent of the unstable range. Therefore, if the instability zone correspond-
ing to the second vibration overtone is higher than the operating ipm of the
rotor, then the ipm margin ijp to the lower boundary of the instability zone de-
338
creases diaring vibrations on lift-off, and it may happen that, at a sufficiently-
large vi"bration anplitude, the lower boundary of the instability zone "descends"
to the operating rpm.
Thus, a helicopter which, in the presence of small vibrations, has an in-
stability zone located above the operating ipm will be stable only at small vi-
bration amplitudes not exceeding a certain critical anplitude a^'^ , which can be
designated as the excitation threshold at vibrations on lift-off.
It is obvious from the above statements that the magnitude of the excita-
tion threshold is smaller, the weaker the forces forcing the tire against the
ground, i.e., the more closely the rotor thrust approaches the helicopter weight.
Thus, the most dangerous situation occurs at the instant of lift-off of the heli-
copter and iirmediately after landing. Consequently, on occurrence of vibrations
dioring takeoff or landing the rotor thrust most be reduced immediately. This
causes the shock struts to operate, inhibiting vibration of bouncing.
It is inportant to note that vibrations on lift-off are dangerous only if
the unstable range is above the operating rpm of the rotor. From this viewpoint,
the landing gear configuration proposed by the Bristol Conpany (see Pig. 3 -IB) is
of interest* As indicated above (Sect. 2, Subsect.2), it is possible in this
landing gear configuration to cause the flexural center of the shock absorber
system to coincide with the center of gravity of the helicopter by choosing the
stiffness of a special spring Cgp if the landing characteristics of the gear are
otherwise satisfactory. Here the lateral forward vibrations of the helicopter
and the angular vibrations about the principal longitudinal axis of inertia of
the fuselage become independent.
Calculations show that in this case the frequency of lateral forward vibra-
tions is approximately the same (somewhat lower) as the frequency of the first
vibration overtone of a helicopter with a landing gear of conventional configura-
tion, whereas the angular vibration frequency may be appreciably reduced in com-
parison with the frequency of the second overtone for conventional landing gears
(in this case, it can even be made equal to the frequency of the first overtone).
Thus, the use of a landing gear of the "Bristol" type permits obtaining /307
a rather low frequency of the second vibration overtone so that the correspond-
ing unstable range will come to lie below the rotor operating xpm. Ground reso-
nance on breaking contact with the ground cannot occur in such a helicopter.
Section 5 • ^ou^i. Resonance, of Helicopt ers of Other Configurations
1. General Comments
As indicated above (Sect .2, Subsect.l), a calculation of natural vibrations
of a helicopter on the ground must be based on the problem of vibrations of a
solid body (disregarding fuselage elasticity) on an elastic base. A solid body
on an elastic base has six degrees of freedom. Accordingly there are six natu-
ral vibration overtones for such a system, each corresponding to a certain vibra-
tion frequency and mode. For a single-rotor helicopter with a slender fuselage,
we were able to consider only lateral vibrations and to disregard yawing oscil-
lations (Sect .2, Subsect .1) .
339
For a helicopter for which the moments of inertia of the fuselage relative
to the three principal axes of inertia are magnitudes of the same oi*der, such a
sinpHfication is iirpermissi"ble« However, if there is a plane of symmetry of
the fuselage, then the longitudinal and lateral vibrations can be considered as
independent. In this case, the calculation of lateral vibrations must take
three degrees of freedom into consideration:
1) lateral displacement;
2) angle of roll;
3) angle of yaw.
A calculation of vibrations in the plane of symmetry (longitudinal vibra-
tions) must also allow for three degrees of freedom:
1) longitudinal displacement;
2) vertical displacement;
3) angle of pitch.
From the viewpoint of ground resonance, both lateral and longitudinal vi-
brations are dangerous .
Below, we will give methods of calculating the natural vibrations of a heli-
copter which take into account all of the above-indicated degrees of freedom.
It should be noted that these methods are applicable also to a single-rotor
helicopter and permit obtaining results more accurate than the results of an ap-
proximate calculation by the method presented in Section 2»
We will also describe a method of calculating ground resonance in air,
caused by elasticity of the fuselage.
2. Calculation of lateral Natural Vibrations with Consideration
of Three Degrees of Freedom
Figure 3 •42 shows a helicopter on an elastic landing gear. Let us choose a
rectangular fixed coordinate system cxyz with its origin at the center of gravi-
ty c of the helicopter. The cx-axis is directed forward (in the plane of sym-
metry of the fuselage) parallel to the surface of the ground, the cy-axis is
directed -upward, and the cz-axis is directed to the right, viewed in direction
of the cx-axis. Let z be the displacement of the center of gravity of the heli-
copter in direction of the cz-axis, and let cpx and cpy be the angles of rotation
of the fuselage relative to the ex- and cy-axes (9^ t»eing the angle of roll and
cp y the angle of yaw) .
The equations of lateral vibrations of the helicopter can be written in /308
the form
^if^y-hy^x=^h\
(5.1)
340
where
m = mass of the helicopter;
Ix and ly = moments of inertia of the fuselage relative to the ex- and
cy-axes ;
I^y = corresponding centrifugal moment of inertia;
Mx and My = moments of external forces acting on the fuselage relative to
the ex- and cy-axes;
Z = projection of the external forces acting on the fuselage at
the c2-axis«
yo\ \^
Fig .3 •42 Scheme of Helicopter on an Elastic
Landing Gear.
Let us first study vibrations in the absence of danping. In this case, the
quantities M^ , My ajrad Z in the presence of small natural helicopter vibrations
relative to the position of equilibrium can be linearly expressed in terms of
the displacements z, cpx, and cpy We can then write the expressions for displace-
ments of the flexural centers of shock absorption (c#fl) in the cross sections
I-I and II-II of the fuselage (RLg.3»42) corresponding to the fore and aft land-
ing gear in terms of the quantities z, cp^* 9y *
1) fore landing gear:
2) aft landing gear:
Z\=^z—^yli—^xeu
Zi^Z-T- i^yl2 9x^2,
341
where
t-L and Ig = distances of the planes of the fore and aft landing gears
from the center of gravity c;
e^ and eg = distances from the cx-axis to the flexural centers of the
fore and aft landing gears.
Knowing the displacement of the flexural center of the fuselage cross sec-
tion 2 in the plane of the given landing gear and the rotation of this cross
section relative to the flexural center (which for "both cross sections will be /308
equal to cp^), we can determine the elastic forces and moments acting on the fuse-
lage in this cross section in the same manner as before (Sect.2> SulDsect.2) for
a plane body on a flexible support. Determining then the quantities M^, My,
and Z, we obtain the following expressions:
(5.2)
where the corresponding stiffness coefficients are determined by the formulas
■c, ei
(5.3)
The quantities c^ , *^zp^ ^cp 9 ^^^ *^cp represent the coefficients of lateral
and angular stiffness of the fore and aft landing gears .
The first equation of the system i,5*2) also contains the term Gz represent-
ing the moment of the force of the helicopter weight G relative to the cx-axis,
generated during the lateral displacement z.
Substituting eqs.(5.2) into eqs.(5*l), we finally obtain the following equa-
tions of small lateral helicopter vibrations:
^y'^y " ^xy ?x = <^i^ - c^iix - % ?y ;
mz==-c^z + c^^^~^c^'^y.
Seeking the solution of this system in the fonn
(5.4)
342
where zq, cp°, cpy, and p are constants, we arrive at the following system of
linear alge'braic equations for determining these constants:
[c, - mp'') z^ - cy^ - r,cp; - 0;
- (G + c,) z, + (c,^ - /^p2) cp; 4- (c^^ -j- /^^p2) cp^ _ 0;
(5.5)
Equating to zero the deterndnant of this system
= U
and perforndng simple transformations, we obtain the following characteristic /310
equation for determining the natural lateral vibration frequencies p of the heli-
copter:
where
A=hjt^-
1;
C=pLp'-pIpI
xu^ yx
n2
^ zti r' It
XIJ'
2/n2
zx^ yz^ xy
— /?2 /?2 ^2 ^ p2p2 p2
zyf^yx^G
PlxPl/l-PlxPlzPlj.
(5.6)
(5.7)
I I
hx = J^^ ; hy = J^^ ' are dimensionless coefficients,
-^x ly
The partial frequencies p^, Pcp > Pep ^^^ "the quantities Pxy* Pyz* etc. are
obtained by means of the formulas
^2_I^ 2 ^^
«2 ^y
m
xy T
P^ =
^zy
^^" ly '
„2 _ _^ .
(5.8)
343
Pi-
m
G
"XZ T *
Equation (5»6) is an equation of the third power with respect to the quanr-
tity p^. It can iDe demonstrated that its roots p^ (k = 1^ 2, 3) are always real
and positive • Therefore, one of the possi^ble methods of finding the natural
vibration frequencies p^ is the graphic method in which we construct the graph
of the left-side of this equation, which is regarded as a function of the quan^
tity p# The points of intersection of this graph with the abscissa give the
values of the natural frequencies (Fig .3 .43).
Let us number the natural vibration frequencies of the system in increasing
order: Pi < Ps < Ps* Let us call the quantities px, Pss Ps^ the frequencies of
the first, second, and third overtones of the helicopter natioral lateral vibra-
tions. Each natural vibration frequency corresponds to a certain vibration mode
characterized by a certain correlation of the anplitudes Zq, cpx> cpy, which can
be found for a given p^ (k = 1, 2, 3) from eqs.(5^5) if we substitute there -py.
for the quantity p . This will yield the expression
(?")*=
(^^)*=
(c
'"Pl){<=fj,'-^ypl)-<''^i^
{c^ — mp\) {cei + IxvpI) — «i««
' [cti -tIxi/pI) ci - (c - lypl) c^
(5.9)
where k =1, 2, 3*
It is easy to show that the vibration mode of a given overtone is char- /gll
acterized by a certain straight Une lying in the plane of symmetry of the fuse-
lage xcy and representing the locus of the points
(belonging to the fuselage) remaining stationary dur-
^ A=A(.p)=/?p*+BpSCp2-hi> j_^ vibrations of this overtone.
In fact, the displacement Za of a certain fuse-
lage point A lying in the plane xcy and having the
coordinates x and y (see Fig .3 .42) can obviously be
determined by the formula
Pj P
Fig .3 .43 Character of
the Graph A = A(p) for
Determination of the
Natural Vibration Fre-
quencies of a Heli-
copter on an Elastic
Landing Gear.
In the presence of vibrations of the k-th over-
tone, we have
z = Zq cos Pf^t\
9y=9l^0spkt
3hU
Hence,
^A = (^0 + 9> - 9;-^) cos p^t =
= ^0 [ 1 + (?;)jfe i/ - (?;);^ ^] cos /7^/.
Therefore, the condition
1 + (9^W-Qa-^^0
(5.10)
represents the equation of the locus of the points in the plane xcy, whose vi-
bration airplLtudes are equal to zero dxoring vibrations of the k-th overtone.
However, this is an equation of some straight line.
-- Thus, the vibration mode of
the k-th overtone can be charac-
terized by the position of some
straight line in the plane xcy.
This straight line will be desig-
nated here as the line of the
nodes of the k-th overtone of
lateral vibrations. The equation
of the nodal line [eq.(5.lO)] is
easily derived by means of eq.(5.9)
for a given value of pj^.
The results of natural lateral
vibration analyses of a helicopter
are conveniently represented as a
sketch giving a side view of the
helicopter and the nodal lines of
all three vibration overtones, with
an indication of the correspond-
ing frequencies (Fig.3*A4).
irsttribr^liS^L^^
Fig .3.244 Characteristic Arrangement of
the Nodal lines of Vibrations of the
First, Second, and Third Overtones.
The approximate calculation method for natural lateral vibrations given in
Section 2 and based on the assunption of independence of yawing vibrations, y^3l2
can be obtained as a particular case of the equations derived here.
If the ex- and cy-axes are the principal axes of inertia (I^^y = O) and if
the conditions
are satisfied, then eqs.(5»4) are resolved into two independent systems of equa-
tions :
, (5.10')
345
Equation (5.10^0 determines the independent yavdng vibrations, while the
system of equations (5»10') determines the lateral vibrations corresponding to
the physical picture presented in Section 2 (in this case, two of the nodal
lines are parallel to the cx-axis, and the third coincides with the cy-axis).
For a real helicopter, the conditions ci = Cgi =0 and I^y = are never
accurately satisfied. However, for helicopters with an elongated fuselage, if
the angle of "between the principal axis of inertia cxq and the cx-axis is low
(see Fig. 3 -42) and the moment of inertia I^ is small in comparison with the two
others (ly and I^), the results of the "exact" and approximate calculations may
agree with an accuracy sufficient for practical piorposes.
To determine the danping coefficients of natioral vibrations, one can use an
approximate method analogous to that presented in Section 2 (Subsect.4) for a
system with two degrees of freedom. For each natural vibration overtone, we
then determine the danping coefficient on the assurq^tion that in the presence of
dajiping the vibrations of this overtone represent also angular vibrations about
the nodal line of this overtone. In this case, just as before (Sect .2, Sub-
sect.4), the equation of natural angular vibrations of the helicopter about the
nodal line can be written in the foiTn
/.? + ^.,? + S9=0, (5.11)
where
Ifc = moment of inertia of the helicopter relative to the nodal
Hne of the k-th overtone;
Geo, ~ Pk^k ~ angular stiffness of the shock absorber system during rotation
relative to the nodal line of the k-th overtone;
kcp = corresponding danping coefficient.
The moment of inertia of the helicopter relative to the nodal line can be
determined by means of the formula:
4 = w/^| + /vC0s2Y,+ /^sin2v;,+ /^^sin2Y;„ (5-12)
where
hjc = distance from the center of gravity of the helicopter to the nodal
line I
Yjj = angle made by the nodal line with the cx-axis (Fig.3.Zf4).
The quantities h^ and y^ are determined from the formulas
tan,,= J^., (5.13)
/(a+(a
346
(5.14)
The coefficient of angular danping kq, is determined from the expression /313
where
^ik
and d
2k
a-i and a-
K^i
and kv
= distances l^etween the nodal lines (Fig -3 '45) and the
lines connecting the point of contact with the ground
of the tires of the fore and aft landing gears;
= wheel tracks of the fore and aft landing gears
(RLg.3.16);
= danping coefficients of the lateral and vertical
springs (see Fig.3»l6) of the fore and aft landing
gear, having the same meaning as in Section 2 ( Sub-
sect •4) •
After determining the quantity
ken , we can derive the dimension^
less coefficient of dairping of the
k~th overtone:
(5.16)
•^Jod*
3. Calculation of Natural Helicopter
Vihrations in the Plane of
Symmetry ( Longitudinal Vibrations)
Fig .3 '45 For Basic Correlations during
Vibrations of a Helicopter Relative to
the Nodal line of the k-th Vibration
Overtone .
Let us tiorn to Fig. 3 -46. The
problem of helicopter vibrations in
the plane of symmetry reduces to
an investigation of oscillations of
a clanped plane elastic solid body
in its own plane {yDj)* The vertical springs with a stiffness coefficient Cy-j^
and Cyg simulate the vertical rigidity of the fore and aft landing gears, while
the horizontal springs c^^^ ^^ ^^^ simulate the fore and aft landing gears in
the direction of the Ox-axis • If the tires of the landing gears are not braked,
then.Cx and Cx^ = 0. In the case of braked tires, the elasticity of the land-
ing gear in the direction of the Ox-axis is conposed of the elasticity of the
tire and the elasticity of the tire suspension system (for exairple, flexural
elasticity of the landing gear struts, etc.)* Fo^ approximate calculations, the
longitudinal stiffness of one tire c^^ can be taken as equal to c^^ 7^ 1.5 Cy^.
Let the point M (Fig.3.Zp6) with the coordinates e^ and ey represent the
flexural center of the shock absorber system at longitudinal vibrations. The
quantity ey is the distance of the center of gravity of the helicopter from the
ground surface, while the quantity e^ is -determined from the expression
_ 'y.^^-
(5.17)
C 4- c
347
let X and y represent displacements of the center of gravity of the heli- /3V1
copter in the direction of the Ox- and Oy-axes, and let cp^ be the angle of rota-
tion of the fuselage relative to the Oz-axis« Then, the equations of small vi-
"brations of the helicopter in the plane :i£fj in the absence of damping have the
form
where
^y = ^^1 ~T^yt\ \
(5.18)
(5.19)
Let us introduce the following notations :
^0 — ^x ^y ~h ^y^x ~^ ^fJ
"* m
. <^y . ;.2^£o^.
m ^ Iz
Q--
Q
e
0+— )
(5^20)
Let us also substitute cp^ by the new variable
s = Q?z-
Equations (5*18) can then be written in the form
(5.21)
x=~plX'^pleyS;
y
s
'^-ply-pl^xSi
: — n2,
Pl^ + PxK^~Ple,y.
Seeking the solution of this system of equations in the form
x=^Xo COS pt; y—ijo COS pi; s-=Sq cos pt,
(5-22)
(5.23)
we arrive at the following system of Unear homogeneous algebraic equations for
deteraiining the quantities Xq, Jq, and Sq:
{pl-~P^)Xo~-plIySo = Q;
iPl~P')yo+Ple,So=Q;
-P^elxo+ple,yo+{pl-p^)so=0.
(5.2ff.)
31^8
ir
Equating the determinant of this system to zero
Z2ii
0;
1^ 772.
P>-^
pI^x
rp. rp
=0,
we obtain the following characteristic equation for determining the natural vi-
bration frequencies p:
where
^=P%^'y-^P9^~PlP\-P\P%-P\Pl>
' = - PlPl {Pl^y^'l + Pi el).
(5.25)
(5.26)
This equation has three real roots p^ which can be found graphically "by
constructing the graph of the function A = A(p) = p^ + ap'^ + "bp^ + c, siinilar to
that indicated in Section 5 ( Sub-
sect. 2) for eq.(5*6) (see Fig. 3*43).
Let us then arrange the roots of
the eq.(5.25) in ascending order
P 1 ^ P3 *^ Pa ^nd designate by the
quantities p x^ Ps^ ^-^d pa the fre-
quencies of the first, second, and
third natural vibration overtones
of the helicopter in the plane of
symmetry, or longitudinal vibra-
tions. To each longitudinal vibra-
tion overtone there corresponds its
own vibration mode of the heli-
copter, which is conveniently char-
acterized by the position of the
corresponding vibration node Oj^
(here k = 1, 2, 3) in the plane :idJy,
i.e., the fuselage points that remain stationary during vibrations of this over-
tone. The coordinates of the vibration node x^ and j^ can be found in the fol-
lowing manner: The vibration amplitude a^ and ay of any fuselage point with the
coordinates Xj^ and j^ in directions of the Ox- and Oy-axes are determined by the
obvious formulas
Fig.3.Z|j6 Scheme of Helicopter on an
Elastic landing Gear, for Calculating
Vibrations in the Plane of Symmetry.
where cpo = SqP ^s the angular vibration aaplLtude of the helicopter.
349
The coordinates x^^ and y^ are determined from the conditions a^ = and
ay = 0, such that
_ yo
Xu^~
?0
^0
<P0 ^0
The values of the ratio
yo
and
Xr
can "be found from the first two equa-
tions of the system (5»2U), if the vibration frequency p is known. For vibra-
tions of the k-th overtone, we obtain
pI'^x
;2 2
Hence, we obtain the following formulas for determining the coordinates /316
of the vibration node:
x.=^-
yk~-
EJl
Py '
ey
1-
Px
(5.27)
We will give two possibilities for a sin^lified calculation of natural heli-
copter vibrations in the plane of symmetry.
When the flexural center of shock absorption M (see Fig.3*46) lies on the
Oy-axis (e^ = O), the equations of motion (5«18) are sinplified and take the
form
/.?.=
(5.29)
Equation (5*28) describes vertical forward vibrations of the helicopter,
which are not of interest from the viewpoint of ground resonance.
Equations (5-29) express longitudinal vibrations of the helicopter, which
in this case can be regarded as a system with two degrees of freedom x and cp^ •
Such a system is mechanically equivalent to the system discussed in Section 2
350
(Subsect.S) and depicted in Fig .3 •16. Therefore, in calculating the natural
frequencies of a helicopter, it is here possible (neglecting the moment due to
the force of the weight G) to use the graphs in Figs .3 ^l? and 3-20 as well as
eqs*(2.22), (2.23), and (2*2i^), putting there
^==^' (5.30)
c.el ^ c^el ' (5.31)
The quantity a^^ = will represent the relative distance "between the vi-
"bration node of the k-th overtone (which here comes to lie on the Oy-axis) and
the center of gravity of the helicopter •
For a real helicopter, the quantity e^ is generally not equal to zero, but
usually is small in conparison with the quantity ^ 1 + ^2 • IJi most cases, an ap-
proximate calculation in which we set ex = will give natural vibration values
close to those obtained by an exact calculation and can be successfully used as
a preliminary calculation whenever one wishes to obtain results quickly, with-
out the need for greater accuracy.
When calculations of longitudinal vibrations are carried out in the pres-
ence of unbraked tires (c^ = O), the equations of motion (5 •18) again are re-
solved into two independent systems:
h^z = - ^0?^ ^-Ox — cye^y .
In this case, we can assume x = during vibrations since there is no /317
projection of the external forces onto the Ox-axis. One of the natural fre-
quencies of the system is equal to zero and corresponds to uniform motion of the
center of gravity of the helicopter along the Ox-axis. The two other natural
vibration frequencies, as in the preceding case, can be found from the graphs in
Figs. 3^19 and 3.20 or from eqs.(2.22), (2.23), and (2.2^) in which we must put
Jz_
4
'^•— ^; (5^32)
met
'- ^^ ? ■• (5.33)
cye
X
X
The values of aj^ = — — represent the relative distances between the center
of gravity of the helicopter and the vibration nodes which, in this case. He on
the Ox-axis.
351
Finally, when there is no tire traking "but e^ = 0, the sinplest formulas
for natural frequencies in the plane of symmetry are obtained:
y m
Pz
To determine the damping coefficients of natural longitudinal vibrations
we can again use an approximate method based on the assunption that, in the
presence of dairping forces, the vibrations of the given overtone are angular vi-
brations relative to the nodal line of the given overtone which, in this case,
represents a straight line parallel to the Oz-axis and intersecting the plane xOy
at a point with the coordinates x^ and j^ [see eqs#(5«27)]» The equation of vi-
brations of this overtone can be rewritten in the form of eq.(5.1l), except that
the quantity Ij^ is found from the formula
I, = I.-Vm{x\+yl). (5.34)
In determining the danping coefficient k^j there is no need to allow for
danping of the longitudinal elastic elements c^^ and c^^ (see Fig ,3 •46) so that
only danping of the vertical elastic elements with stiffnesses c^ and c^ of
the fore and aft landing gears must be considered (see Fig. 3. 16).
The corresponding dairping coefficients ky^ and k^^ are determined as indi-
cated in Section 2 (Subsects.4 and 5)»
By calculating the moment from the dajrping forces relative to the nodal
line, we obtain the following expression for determining the quantity ]<i^ :
K =2 [^;. {h-x,y+k'^^ {k+xuf]. (5.35)
The dimensionless danping coefficient H^ of the given vibration overtone is
determined by the formula
4. Reduction of the Problem to Calculation o f a Rotor /318
on an Elastic Base
After determining the natural vibration frequencies and modes of the heli-
copter on an elastic landing gear, the calculation of ground resonance can be
352
reduced to the calculation of a rotor on a flexible support, as presented in
Section 1.
The method of calculation "based on reducing the problem to a rotor on a
flexible sipport is an approximate method and analogous to that given in Sec-
tion 2 (Subsect#6) for a single-rotor helicopter.
The essence of the approximate method is as follows: A separate calcula-
tion of ground resonance is performed for each natural vibration overtone; here,
the helicopter fuselage is regarded as a solid body -with one degree of freedom,
namely rotation about the nodal line of the given overtone. Of coiorse, such an
approximate method holds only for the case in which the natioral vibration fre-
quencies of different overtones are sufficiently "far" from each other.
When there are two "close" natural vibration frequencies, certain correc-
tions must be introduced into the calculation. The method of refining the cal-
culation will be presented below.
Thus, for calculating ground resonance, the helicopter vibrations with re-
spect to each overtone are separately considered as angular vibrations of the
fuselage about some fixed straight line: nodal line of the given overtone.
It can be demonstrated that, with such a sinplification, the equations of
motion of the system reduce to a system of equations analogous to the system
(1.16) (Sect.l). in this case, all formulas of Section 1 remain in force and
we can use the graphs for determining the instability fringe (see I^gs.3'3 to
3. 12); however, here the quantity no is to mean a dimensionless danping coef-
ficient n^ of the given vibration overtone determinable from eqs.(5*16; or (5.36)
(Sect. 5, Subsects.2 and 3), while the quantity e is to mean the quantity e^^ cal-
culated for the given overtone by the formull^
where
i=l, 2f ..., s;
s = n-umber of rotors, with each of the quantities e^^, determined by the
formula
\ 2 Iy,f, Ik 1 1
Here,
Ifc = moment of inertia of the fuselage (with the rotor masses concen-
trated at the center) relative to the nodal line of the k-th over-
tone [see eqs.(5»l2) and (5*34)];
l^ = distance between the center of the given i-th rotor and the nodal
Une of the k-th overtone if lateral vibrations are considered, or
the distance between the nodal line of the k-th overtone and the
plane of rotation of the given rotor if longitudinal vibrations of
353
the helicopter are considered;
n = mjinber of "blades of a given rotor;
Sy^^ and ly, jj = static moment and moment of inertia of the rotor "blade
relative to the drag (vertical) hinge.
The rotors can "be different; however, the a"bove method is vaUd only if all
rotors have identical angular velocities of rotation and identical values of the
parameter Vq [see eq.(l.9)]»
As indicated above, the approximate calculation method presented here /319
holds only if the vibration frequencies of different overtones are sufficiently
"far apart". It can be demonstrated that, if there are two close natural lat-
eral (or longitudinal) vibration frequencies - for exanple, p^ and Pn, - then the
calculation of the boundaries of the instability zones can be performed for one__^
overtone - for exanple, p^ - but is refined by substituting a certain quantity n^q
for njjj (for a given overtone, where n^^ < n^), which is determined by the formula
- 1
'^ '"1+^^ * (5-39)
This formula is derived for the case of p^ = Pn^ i.e., when the natural vi-
bration frequencies of the two overtones in question coincide exactly. J£ p^ ^
^ p^, then eq.(5.39) yields an understated value of Uq^^
If there are two close natural vibration frequencies p^ and p^, with one of
them - for exanple, Pj^ - being the frequency of the nr-th overtone of lateral
vibrations and the other, p^j, being the frequency of the mr-th overtone of longi-
tudinal vibrations, then, generally speaking, a rotor on a flexible sipport with
two degrees of freedom must be considered (see Sect.l, Subsect.4). In this
case, it is possible to approximately estimate (within the safety factor) the
required dairping by eq.(1.52), for a rotor on an isotropic flexible sipport, sub-
stituting into it the quantities no and e for that of the two examined over-
tones for which the value of the ratio is smaller.
It should be noted that such a calculation is required only in the rather
rare case in which, for both examined overtones, not only the values of the fre-
quencies Pn and Pm but also the values of — ^ and — ^ are close* If the quan-
e.
n.
■k
tity for one of the overtones is larger by a factor of 2.5 " 3 than for the
other - for exairple, = 3 • then we can disregard vibrations of the
n-th overtone, and examine only vibrations of the m-th overtone (as independent).
5 . Self-Excited Vibrations in Flight of a Helicopter with
an Elastic Fuselage
Self-excited vibrations of the ground resonance type are also possible in
354
helicopter flight. The fuselage of a real helicopter is an elastic system which
has its own natural vibration frequencies and modes. If the vihration mode of
ar^ overtone of an elastic fuselage is such that the center of the rotor (or
centers of the rotors) during vilDrations of this overtone is displaced in the
plane of rotation of the rotor, then ground resonance is possible and the fuse-
lage will execute vibrations with the mode of this overtone.
The natural vibration frequencies of an elastic fuselage are usually high
in conparison with the vibration frequencies of a helicopter with shock absorp-
tion of the landing gear, and only one or
two low natural vibration harmonics are
dangerous from the viewpoint of the pos-
sibility of self-excited vibrations.
The lower natural vibration frequencies
of the fuselage usually correspond to its
f lexural vibrations .
Figure 3*47 shows the vibration /320
mode of the first partial of bending of the
fuselage of a Mi-4 helicopter in the hori-
zontal plane. The vibration mode is given
as a curve of the elastic line u = u(x)
(u being the vibration airplitude of the
point with the coordinate x) .
Fig. 3 .47 Mode of First Vibra-
tion Overtone of an Elastic
Helicopter Fuselage.
The natioral flexural vibration fre-
quencies and modes of a fuselage can be
found by conventional methods used for elas-
tic beams of variable cross section (see,
for exanple. Chapter II of this volume) or can be determined experimentally (if
a full-scale helicopter is available) .
If the frequency po and mode u(x) of any flexural vibration overtone of the
fuselage are known, the calculation of self-excited vibrations with the mode of
this partial can be reduced to the calculation of a rotor on an elastic base,
using the formulas in Section 1 or the graphs in Figs .3 •3 - 3*12. In this case,
the quantity e should be determined by means of the formula
E^Sj-f £2+ •
^.=y.
(5.40)
where s is the number of rotors.
The quantities e^ (i = 1, 2, ..., s) are determined from
o2
2 h-hrn^
"i^J^ =-^Q[u,{x)]^dx,
(5.41)
(5.42)
355
where
Ui {X) :
U{Xi)
Xi = coordinate of the center of the i-th rotor;
p = linear mass of the fuselage (with the integral taken over the enr-
tire fuselage length).
The quantity Ui(x) represents the vi"bration airplitude at the point with the
coordinate x, referred to the vibration aiiplitude of the center of the i-th
rotor* The quantity m^^ is the maxiiiiuin value of kinetic energy of the fuselage
during vibrations with respect to the mode of the given overtone, with the vi-
bration anplitude at the center of the i-th rotor being equal to unity, referred
to the quantity pf .
The quantity Tlq should then be equal to the dimensionless coefficient of
dairping of the given vibration overtone of the fuselage. It is determined ex-
clusively by Ir^steresis losses in the fuselage design and usually amounts to
0.02 - 0.05.
Such a coirparatively low value of Eq does not permit elijninating ground
resonance in flight by means of a blade danper, and flight safety of the heli-
copter can be ensured only at sufficient
rpm margins up to the lower fringe of ir>-
stability. Consequently, self-excited /321
vibrations in the air are dangerous only
for helicopters with coirparatively low
natural vibration frequencies of the
elastic fuselage. For exanple, for the
Mi-4 helicopter the rpm margin -up to the
lower boundary of instability correspond-
ing to the first vibration overtone of
the fuselage (see Fig.3.47) is 2B%»
^^LJtlTl::!:^
'■^■iL':]j:^4U^''
^^^....ml'^o
Ground resonance in the air consti-
tutes the greatest danger for helicopters
of side-by-side configuration with a long
elastic wing (Pig.3.4S). The danger of
self-excited vibrations for such heli-
copters is aggravated by the fact that
the rotor centers coincide with the anti-
nodes of the corresponding vibration har-
monic, which yields conparatively small values m^^ [eq,(5.42)] and, consequent-
ly, relatively wide instability zones.
Fig. 3 .4s Mode of Lower Vibration
Overtone of a Side-by-Side Heli-
copter, Most Dangerous from the
Viewpoint of Ground Resonance.
Section 6. Selection of Basic Parameters_of landing Gear and Blade
Dampers. Design Recommendations
As indicated by the general theory of stability of a rotor on an elastic
356
■■■■■ ■ ■■■■■ I
■■ ■■ ■ III 1 1 I ■ III
"base, the stability margin, generally speaking, can "be increased by increasing
the degree of blade vibration danping as well as the fuselage 'vibration danping,
i.e., iDj increasing the dairying capacity of the landing gear.
However, the possibilities of increasing these types of danping are quite
limited in practice, since both the blade dairper and the landing gear have a
number of other functions not related with ground resonance.
The blade danper works in forward flight of the helicopter and loads the
blade root with a variable bending moment which is greater the greater the de-
gree of its danping. The mechanical strength of the root portions of the blade
and hub, and consequently their weight, is determined mainly by the presence of
a danper*
An extreme increase of the degree of dairping of the landing gear without
the use of special devices leads to an increase in shock absorber stiffness and
hence to an increase in the dynamic loads during landing of the craft.
These aspects of the work of blade danpers and of the landing gear must be
considered in designing a helicopter. It frequently is iirpossible to provide a
sufficient margin with respect to ground resonance without using special devices,
either in the blade dairper or in the landing gear system.
For helicopters of single-rotor and fore-and-aft configuration, ground reso-
nance during the ground run may prove the most dangerous. Therefore, this is
conveniently considered to be the design case for selecting the parameters of
blade and landing gear danping. For simplicity, we can consider that the heli-
copter oscillates about the horizontal axis going through its center of gravity,
which is a sufficiently valid assimption at high taxiing speed(Sect.4,Subsect .2) .
As shown above, we derived very sinple calculation formulas [eqs.(A-*lS)-(4»2l)]
for this case and were able to determine the required characteristics of land- /322
ing gear and blade danper by the sinplest method. However, after having selected
the parameters for landing gear and blade danpers, a conplete calculation of
ground resonance for all possible cases is required, including ground resonance
during the ground run, followed by plotting a diagram of safe rpm (see Fig. 3. 25)*
If necessary, the selected characteristics of the landing gear and hub can then
be corrected -
1. Selection of Blade Damper Characteristics
The main characteristic of the work of a blade dairper is the fact that the
natural blade frequency (characteristic frequency for ground resonance) is al-
ways by a factor of about 3-4 lower than the frequency of forced blade vibra-
tions in forward flight.
In fact, in flight a blade executes forced vibrations relative to the flap-
ping and drag hinges with a frequency o) equal to the rotor rpm whereas the natu-
ral blade vibration frequency is p^ = VqCD. Usually, Vq = 0.25 - 0.3; in any
case, the angular ^velocity uo of rotor rotation at ground resonance cannot be
greater than the angular velocity of rotor rotation in flight.
357
This characteristic explains, in particular, the unsiiit ability of using
danpers with a linear characteristic ( Sect #3, SulDsect.3) in view of the fact
that a Unear danper, at constant vibration anpHtude, will generate a moment
proportional to the vibration frequency.
The siirplest danpers producing a moment independent of the vibration fre-
quency are friction danpers and hydraulic danpers with stepped characteristic,
where this characteristic should be as close as possible to the characteristic
of the friction danper (see Fig.3*31^b), A stepped hydraulic danper of this type
is suitable for heavy helicopters since it is lighter in weight than a similar
friction danper, the gain in weight of the danper increasing with an increase in
its power.
When using ordinary danpers, the moment Mq of the danper is selected from
blade strength considerations, while its danping coefficient is determined from
eq.(3»23). Here, the danping margin for ground resonance can be ensiored only
by proper selection of the landing gear characteristics* When this is inpos-
sible, special designs of blade danpers might be needed, which would produce
large blade danping at low vibration frequencies (characteristic for ground reso-
nance) and small blade danping at vibration frequencies corresponding to heli-
copter flight. One of the sinplest types of such a danper is one connected in
series with an elastic element (see Pig .3 .33)- Figiu:»e 3*49 shows one of the
possible design versions of such a danper. let us designate this type of danper
a "spring dajiper""''".
^^m 77777/^/// ////fUm^
Pig .3 .49 Danper with Series-Connected Elastic Element.
1 - Elastic elements (rubber); 2 - Casing; 3 - Safety
valve; 4 - Roci; 5 - Adjusting needle.
To estimate the advantages of a spring danper, we will conpare it with a
conventional friction danper. Let the helicopter undergo ground resonance dur-
ing the ground run so that the center of the instability zone coincides with the
operating rpm of the rotor. Furthermore, let the maximum moment in flight,
permissible with respect to strength considerations of the blade, be equal to Mq .
'"* The design of a spring blade danper for eUbninati-ng ground resonance was pro-
posed by engineers O.P.Bakhov, L.N.Grodko, I.V.Kurova, and M.A.Leykand (Patent
No. 184342).
35^
Then the equivalent danping coefficient with a friction danper is determined /323
by the following formula (3 •23):
Ufrici _A ^
where 5 i is the anplitude of the first harmonic of l^lade vibrations in the plane
of rotation •
When using a spring dairper, the corresponding equivalent danping coeffi-
cient is determined from eq.(3*17)
'' 1+r^^'
■m
where Pb is the frequency of blade vibrations at ground resonance, which can be
considered equal to the product VqUj.
The moment produced by the spring damper in flight can be determined by
means of the formula
which, in the presence of harmonic blade vibrations with a frequency uo, gives
the following value of the anplitude of the moment M [see eqs.(3»17)-l :
M- *"^'
/-(vr
(6.1)
let us now pose the following question: If we select the values of c and k
for a spring dairper such that it produces in flight the same moment Mq as the
friction danper, then what is the maximum value of k^q^^^^ obtainable by varying
the quantities c and k? Here, we will consider that the anplitude of blade vi-
brations with respect to the first harmonic §i in fUght and during the ground
run of the helicopter is the same.
The relative increase in datrping when using a spring danper is conveniently
characterized by the quantity /32^
^trT"^ « -h
M;"' 2 Mo ^^IhELY
(6.2)
359
Substitirting in this formula p^, = v^w and taking into account the condition
M = Mq, we obtain
^ 2 ■l+vp2 »
(6.3)
where the dimensionless quantity is
k=^
(6.4)
Thus, the relative advantage gained_from using the spring danper depends
exclusively vipon selecting the value of k.
Vo-0.15
1
^ 1 l^ivo^)"-
v
'V
s
A
f
\
V
'^
*^
1
,
2^6
W 1Z n W
R-^
Fig .3 -50 Dependence ^ = f(k) for
vo = 0.25-
Figure 3 -SO^ gives a graph of the de-
pendence t(k} for the case Vq =
= 0.25* As we see from this graph,
an increase in k causes the quan-
tity \|f to increase first and then to
decrease, attaining a maxmum j[_ =
at a certain value k = k
opt
which we will call optimal.
Equating to zero the derivative
dk
we find
^opt
2 voVl-vl
(6.5)
(6.6)
At Vo = 0.25, we olDtain kopt = 3-74, Kax = 3.24*
Thus, the use of a spring damper permits increasing the danping at ground
resonance l::y a factor of more than 3, while keeping the moment loading the "blade
in fHght constant •
However, this does not exhaust the advantage of a spring dairper as comr-
pared to a customary danper. In fact, a spring dairper gives "elasticity" in the
drag hinge (c^q), and the presence of such elasticity, as is shown in Section 1,
Subsection 2, reduces the extent of the necessary danping [see eq.(1.3l) and /325
the graph in Fig.3.13].
Calculations show that with consideration of all above statements, the
danping margin at ground resonance can "be increased by a factor of 5 - 6 while
keeping unchanged the moment acting on the blade in flight.
360
f
2. Rotor with Interblade Elastic Elements and Dampers
So far we discussed only the case where the elastic element and daiiper in
the drag hinge are lashed up between the "blade and the hub casing so that the
moment acting on the blade depends exclusively on the motion of the given blade
and is independent of the motion of the
other blades. Occasionally, hub designs
with so-called interblade coupling are
used. The diagram of such a hub is shown
in Fig ,3 ,51. Let us assume that each such
interblade element has a certain stiff-
ness c and a danping characterized by the
coefficient k, so that the force P acting
on such an element is connected with the
variation of its length s by the relation
dt
Fig .3. 51 Diagram of Rotor Hub
with Interblade Coupling.
In this case; the moment exerted on a
given (k-th) blade by the interblade ele-
ments will depend not only on the motion of
this blade characterized by the angle 5ic("^)
but also on the motions of the two adjacent
blades 5k- iC"*^) ^"^ 5k+i(*t')»
At small vibrations of the blades relative to the drag hinges, the moment
acting on the k-th blade will be expressed by the formula
^-^o(?ft-^.-i) + ^o(^.-^.+i) + ^oa.-^.-i)+^o(^';.-^;.+i)»
whence
(6.7)
where h is the arm of the interblade element (see Eig.3.51).
Therefore, the equations of motion of blades in this case have the follow-
ing form [coH^jare with eq.(1.8)]:
where
^ = 1.2.3.
(6.S)
Ji.
361
If the rotor shaft vibrates harmonically
X = Xq cos pty
we can find the forced vibrations of the "blades. The right-hand sides of
eqs#(6.8) in this case have the form
/326
^0 2^/K..{sm[(/;-o.)/+?^A]+sin[(/. + a.)^+?^^]}
Equations (6.8) then permit a solution of the form
^ft(0 = 5oiSin
n
-f ?02sin
(p + a>)^+?^./e].
Let us calculate the elastic moment exerted on the k-th blade by the inter-
blade elastic elements during blade vibrations with respect to some one of these
harmonics - for exanple, the harmonic (p - o)) = p^ . We have
Moreover^
where
2 2jt .
Using these expressions, we obtain
Taking into account that
sinfcp;^4-— Wsm<pj^cos — + costp;^sin — ;
sin
r* — ;^J=sin9^cos - — cos<?Asm— ,
we finally obtain the following e^^ression:
362
M^i ==2co[l — cos^lEoSin(pjfe=
= 2co h ~ cos ~-j ?o sin U^"^- ^1 .
In the case of ordinary elastic elements of angular stiffness c^^ located
between the blade and hub casing, we would have
^el =^/^^oSin
n
Thus, the interblade elastic elements for the given blade are equivalent to
one ordinary elastic element of stiffness
^tf ^ — -^^0
-cos
'"]■
(6.9)
We can also establish exactly that the interblade danpers for the given /327
blade are equivalent to one ordinary danper lashed v;p between the blade and hub
casing and having a danping coefficient
'«?
-Ikti
1 — cos
2n
(6.10)
Consequently, calculation of ground resonance of a helicopter with elastic
interblade coupling and danpers can be carried out by conventional formulas,
taking the coefficient of the danper as equal to k^^ and the stiffness coeffi-
cient in the drag hinge as equal to c^q .
Table 3*2 presents the values of the
quantity
TABLE 3
.2
Number
of
Blades
2
3
4
5
6
C eg. keq.
4
3.73
2
1.382
1
^0 *o
-cos -
2it
(6.11)
for rotors with a different nijimber of blades.
One of the shortcomings of a rotor with
interblade danpers lies in the fact that,
during simultaneously deflections of the blades relative to the drag hinges (all
to one side and by the same angle) which might occur in transition f Hght regimes
and during run-up of the rotor, such danpers do not operate.
In existing hub designs, this drawback is sometimes eliminated by using
conposite designs in which the elastic elements are made in the form of inter-
blade couplings while the danpers are made separately for each blade, i.e., are
mounted between blade and hub casing.
363
3. Selection of Stiffness and Dajnping Characteristics
for landing Gears' '^'
After choosing the characteristics of the blade dairpers, the basic parame-
ters of the landing gear can be selected. For helicopters of the usual single-
rotor and fore-and-aft configurations the wheel track 2a (see Plg-3.17) should
be selected such that the natural vibration frequency pp^^ of the helicopter dur-
ing the ground run (rotation about the longitudinal axis going through the
center of gravity) with inoperative struts (only the tires are operative) is ap-
proximately 20^ higher than the operating rpm of the rotor. This is given by
the condition (4»19):
^=L2.,^=y'^
Ppn~-
If the landing gear is of the four-wheel type, the quantity 20^"" a^ in the
above formula must be replaced by the quantity 0^3=0^^= o^ [see eq.(5.3)].
Since the tires are selected in terms of a standing load, the quantity c^^
in the given formula can be considered as known; therefore, it will yield the
corresponding value of a.
The stiffness of the shock absorbers and their danping can be selected by
assuming that the center of the instability zone during the ground run (during
vibrations with operative struts) coincides with the operating rpm of the rotor.
Such an approach issues from the following considerations: If the stiffness /328
of the shock absorbers is selected such that the unstable range during the
ground run is greater than the operating rpm, ground resonance might occur at
the instant of becoming airborne (see Sect .4, Subsect.3) since, during vibrations
of the helicopter on lift-off of the tires, the instability zone can "descend"
to the operating rpm. It is usually ijipossible to make the instability zone
lower than the operating rpm (with the exception of the landing gear of the
Bristol system whose design, however, is rather conplex) since this would re-
quire an unfeasibly low stiffness of the shock absorbers. On the other hand, if
the instability zone is located directly at the operating rpm and the danping
margin is sufficient, no ground resonance on breaking contact with the ground
can occur since, during lift-off of the tires, the instability zone will be lower
than the operating ipm. This was checked in numerous calculations and program-
ming on an electronic coiiputer of ground resonance on tire lift-off, performed
by engineer Yu.A.Myagkov.
For siirplicity, let us assume that the landing gear is equipped with verti-
cal shock absorber struts (see Fig.3.l7,b). As shown in Section 2, Subsection 5,
the maximum danping of the tire-oleo system obtainable in choosing the optimal
Cs a
danping of the shock absorber depends on the ratio '- — . Making use of
"''' The method of selecting the landing gear parameters proposed here was developed
by engineer Yu.A.Myagkov.
364
eqs»(2#37) and (2»38) and considering that, diiring the ground run.
^^'- ;,,= ./ ?^«^
we can obtain the following formula which determines the maxLm-um possible coef-
ficient of available helicopter danping during the ground run:
l/^oJmax /^(l_L.^) ' (6.12)
where
'^='^' (6.13)
This means that the maximum possible dairping coefficient which can be ob-
tained during the ground run by varying the quantity kg^ ^ depends exclusively on
the ratio — ^* ^ * Therefore, knowing the danping reqirLred for the elimination
^ pn
of ground resonance, it is easy to determine the necessary stiffness of the
shock absorber Cg.a • ^ "the blade danping is known, the required dairping no
can be determined by eq»(1.31)
where n^^ is the danping coefficient of the blade n^ referred to the natural vi-
bration frequency Pp^ of the helicopter during the ground run with inoperative
struts (using only the tires):
Ppn
/ 2cfa2
(6.15)
p''^" is the natiH'al vibration frequency of a helicopter with operative struts /329
at optimal dairping, referred to the quantity pp^^ :
?- = -^=l/-?^. (6.16)
Ppn K 1+2*
It is required to provide a danping margin of
,_J^oK (6.17)
^^ r 1
365
Using eqs-.(6.l2), (6.14), and (6#1j6), we obtain
0.25
/x(i4-^:
) ^(1-vo) V
cc
n
70
8
6
\
\
\
K
\
V
^
O.Z 0,^ 0.6 0.8 %
Fig. 3. 52 Graph of the
Dependence of the Coef-
ficient a on H .
This relationship can tie rewritten in the fol-
lowing manner:
where
= (l-vo)
V-
«=l/-
2(1 +2x)
*2(H-x)
(6.1S)
(6.19)
After selecting the blade and tire character-
istics and designating the necessary darning
margin T], the left-hand side of eq.(6.18} is known.
Knowing the quantity cv, it is easy to find the quan-
tity K from eq.(6.l9) and then the necessary stiff-
ness Cg.a of the shock absorber- For convenience
of determining n. Fig .3. 52 gives the graph of the
dependence c^(h).
To select the stiffness c^.a ^7 "^^^ indicated
method, we can take T] = 1 since the '^kinematic" dairp-
ing of the tire during the ground run is disregarded in the formulas [see
eq.(4*2l)]« The actual dairping margin T] with consideration of this additional
danping should be at least 1.5 - 2"'»
After the stiffness of the shock absorber is found, its optimal datiping co-
efficient can be determined by eq.(2.36), namely
where
--PpnP
-Pp-\/\
+ 2x "
(6.20)
(6.21)
Since, in reality, the characteristic of the shock absorber danping is gen-
erally nonlinear (Sect .3, Subsect.l), we must understand by the quantity kgj^
the dairping coefficient of an equivalent Hnear shock absorber.
"''" It should be recalled here that the case without kinematic danping is obtained
during vibrations of a heUcopter on ice, when there is no friction between tire
and ground (see Sect .4, Subsect.2).
366
4. Cer tain Recog giend.at ions for landing Gear Design
7330
One of the "basic difficulties in designing a landing gear is the coiiplex-
ity of providing the necessary dairping of the shock strut. If the size of the
orifices through which the hy-
draulic fluid passes when the
shock alDsorber is operative is
selected from the condition of
ground resonance, then, as a
rule, the work of the shock ab-
sorber during landing will be
unsatisfactory (the forces will
be too great when making contact
with the ground). If this size
is selected from the conditions
of landing, then we obtain too
small a dairping during helicopter
lateral vibrations, which is
conpletely insufficient for
avoiding ground resonance.
This difficulty can be over-
come by two methods (Ref .18):
1) increase in dairping
on the backstroke of
the shock absorber;
2) installation of spe-
cial valves in the
design of the shock
absorber.
Instant of valve opening
Oil
Valve opens only on landing at
instant of maximum overload
the
Orifices for damping of
ground resonance
Fig .3. 53 Shock Strut with Valve.
The first of these methods
is the sinplest and involves the
following: The size of the ori-
fices through which the hydraulic
fluid is forced during the for-
ward stroke of the shock absorber
(coirpression) is selected from
the landing conditions, while
the size of the orifices throiogh
which the hydraiolic f l-uid passes dioring the return stroke of the shock absorber
(extension) is selected from the ground resonance conditions. This is possible
because of the fact that, during helicopter vibrations, one of the shock struts
(right or left) executes a backstroke at each instant of time. Therefore, /3 g , l
generally speaking, the necessary danping coefficient of the helicopter at ground
resonance can be seciored only by danping in the backstroke of the shock ab-
sorbers .
However, danping in the backstroke can be increased only within certain
limits. An extreme increase of danping in the backstroke (very small orifices)
leads to a very slow "emergence" of the shock absorber struts from a conpressed
state after touchdown. Therefore, in heavy rolled landing on rough ground when
367
the first touchdown may "be followed ^jy further inpacts, such a method of increas-
ing the danping might be unacceptable.
The second method does not have this shortcoming and involves the follow-
ing: A special spring valve is placed in the shock absorber, which opens only
when the conpressive force in the shock absorber exceeds (at touchdown) a certain
critical value Pg^. a^ ^^ Ps.a < ^s^J^a^ those orifices whose size had been se-
lected from conditions of ground resonance will be operative while at Pg^^ >
> Ps^/a the orifices of larger diameter whose size had been based on conditions
of limiting the landing overload become operative. Figure 3*53 shows a design
scheme and a diagram of dynamic conpression of such a shock absorber.
Another ijrportant factor to be allowed for in designing a landing gear is
the inevitable presence in any shock absorber of prestressing forces (Sect. 2,
Subsect.y), i.e., forces in whose presence the shock absorber begins to operate.
For a helicopter landing gear, it is desirable to have the smallest possible pre-
loading forces Pq since, at high rotor thrust, the forces P on the landing gear
decrease and since, at P < Fq, the shock absorbers do not operate. In this case,
ground resonance may develop with inoperative shock absorbers on elastic tires
which are virtually without danping. For helicopter landing gears, the strut
characteristics must be chosen such that the prestressing force will not be more
than 10^ of the standing load on the shock absorber at zero rotor thrust.
368
CHAPTER IV 7332
THEORETICAL PRINGIPI^ OF CALCULATING BEARINGS
OF MAIM HELICOPTER COMPONENTS
The service life of the main components of a helicopter depends in many re-
spects upon the performance of their iDearing assemblies, which means that con-
siderable attention must be paid to problems of the theory of calculating anti-
friction bearings in helicopter engineering.
As known, the life expectancy of general-purpose antifriction bearings
may vary within wide limits owing to various factors of a metallurgical and tech-
nological nature. In this respect, the necessary reliability of bearing assem-
blies in general machine construction is achieved by introducing suitable safety
factors, i.e., some overestiinate of design loads. It is logical that, in this
case, the requirement for accuracy of calculation of bearings can be reduced sub^
stantially. Of course, for aircraft corrponents, where an increase in reliability
should be attained by iirproving the design without increasing the size and weight
of the bearing assemblies, such a procedure is unacceptable. This is all the
more so since aircraft bearings are manufactured from high-quality materials,
have high precision, and are subjected to very strict inspection in production,
as result of which the dispersion of their service life is noticeably reduced.
Aircraft bearings, including those used in helicopters, should be calculated as
accurately as possible with consideration of the peculiarities of their loading
and service.
In recent years, thanks to studies by Soviet and foreign researchers, con-
siderable advances have been made in practical calculation methods for antifric-
tion bearings; nevertheless, these are by no means always sufficiently accurate.
This is especially true of bearings working under coirplex combinations of ex-
ternal loads and vibrations with small anplitudes; these are the cases of
greatest interest for helicopter engineering. The lack of reliable calculation
methods for antifriction bearings working under the above-indicated conditions,
handicaps the design of reducing gears, pitch controls, and hubs of the main and
tail rotors of helicopters. We can cite many exanples where these vitally im-
portant conponents failed prematurely due to the failure of iirproperly selected
bearings .
In this Chapter, we will attenpt to report the results of theoretical and
experimental investigations which had the purpose of refining the calculation
methods for bearings of helicopter conponents. As shown in practical use, the
methods of calculation given below permit a fuller utilization of the load-carry-
ing capacity of the bearings. Such methods, in designing bearing assemblies, /333
have frequently made it possible to create sufficiently contact and light struc-
tures capable of operating reliably for protracted periods of time at relative-
ly high loads.
369
Section 1# Equations of Static Eqidlibrium of Radial and Radial-Thrust
Ball Bearings under Combined Load
The relations used in the calculation of "bearings are based on results of
investigations of. the distribution of external loads over the rolling bodies #
We will construct equations from which we can derive the pressure on the
balls in the general case of load-
ing of radial and radial-thrust
ball bearings •
Let a single-row ball bearing,
after landing, have a radial play
2 A on the shaft and in the housing
at an established operating tenper-
atiire regime of the assembly.
let us take a rectangular co-
ordinate system xyz with its origin
at the center of the outer race#
The X-axis is directed along the
axis of rotation of this race (see
Fig .4-1).
Upon applying an arbitrary ex-
ternal load to the bearing, the
center of the inner race is shifted
to a point 0' with coordinates s, t, and u, while its axis of rotation x' is de-
flected relative to the x-axis throiogh some angle ■& whose projections onto the
planes xOy and xOz are equal to «^x and i^g, respectively (RLg.4*l)-
Let us assume that the ball whose center O^a ^^b in the plane PI which,
together with the plane xOz, makes the angle ij; is acted ipon by normal forces P|
identical in magnitude and directed along a common straight line passing through
the centers O^^t ^"^ *^in ^^ "*^^® cross sections of the raceways of the outer and
inner races and the point 0^^ (^g»4-2)* As is common in the theory of anti-
friction bearings, we will disregard ar^ displacement of the center of the con-
tact area of the ball with the inner race from the plane PI as well as the tan^
gential forces arising at the points of contact of the ball with the races.
Fig.4»l Scheme of Displacements of In-
ner Race of Bearing under an Arbitrary
External load.
According to the well-known Hertz formula, we have
p^^Blf^
(1.1)
Here, 6^ is the convergence of the raceways of the races in the direction OoutC>in
due to elastic deformations at the contact zones.
For ball bearings with the usual internal geometry, we can pub
5 = v5o-^,,
(1.2)
370
where
V = factor depending on the relation "between the radii
and r^- of the raceways of the outer and inner
^out
g = ^out*** ^In - d
ba
races and the diameter of the ball d^a*
distance "between the points 0^^^ and Oj^ at the mo-
ment of contact of the "ball with the races (when ^^ =
= 0).
If the diameter d^a is expressed in millimeters and the forces in kilo-
grams, then at a modulus of elasticity E = 2.08 x 10^ kg/cm^, of the material of
the races and "balls, the coefficient Bt,
is equal to 62»
9~^~tsin(p + u cos
^l^'gCOSCff
Fig. 4. 2 Polygon of Forces Act-
ing on the Ball.
The factor v has values indicated
in Ta"ble 4«1*
TABLE 4-1
^out(in)l^ba
0.510
0.63
0.515
0,520
V
KOO
1.39
It follows from the conditions of
static equilibriimi of the bearing ele-
ments that the external forces and mo-
ments appHed to the inner race can be
written as (see Fig. 4.2)
/^^ ^= __ "V />^ cos P4, sin 6:
^z = 2 ^4" ^^^ P^ ^^5 ^T^'
My = rQ^P^sin ^^ cos ^;
M^^Tq^ P^ sin '^^ sin 6,
(1.3)
Here,
P^ = angle of contact between ball and races;
ro = radius at which the centers of the balls are located-
extends over all loaded balls.
The sign S
Let us assume that the races have a perfectly regular geometric shape which
does not change when a load is applied. In this case, to determine the conver-
gence of the raceways 6^ and the angle of contact p^ we can use the formulas
371
5^^[(5 + Vosm^ + VoCOS^)2H-(^-A~^sint;> + r£COsO)2]i/2_^; ^^^^^
Plaving e^^Dressed, in eqs#(1.4) and (I.5), all linear quantities in frac- /335
tions of the distance g, we can rewrite them in the forai
84,= [(s + '^iSmO+l2COs6)2 + (cos3o-^sinO + wcos->Pl'^^ (1.6)
cos po — ^sin 4* + ucos<\> '
where Bq = cos""^ is the so-called initial angle of contact (angle of con^
g
tact xn purely a:x:ial displacement of the races due to the operating radial
play 2 a).
■yy
In eqs#(1.6) and (I.7), the terms F^ and 'e^ denote the quantities '^i — —
g
and ??2 — —*
g
It should be borne in mind that the operative axLal play of the bearing Sq
is connected with the angle Po ^7 "t-he following relation:
2^0 = 2^ sin ?o
or, changing to relative quantities,
27o = 2sin3o. (1.8)
The relative quantities are denoted everywhere by the same letters as the
absolute quantities but with vinculi.
These equations describe the conditions of static equilibriiM of radial and
radial-thrust ball bearings under any combinations of external loads* They per-
mit finding all parameters characterizing the distribution of forces between in-
dividual balls. However, it should be remembered that, due to the coirplexity of
the correlations between the quantities 6j and P^ and the relative displacement
of the races, practical application of these equations involves a large calcula-
tion volume. In engineering calculations these are usually replaced by various
approximate correlations. One of the most convenient variants of such correla-
tions, with a sufficiently high accuracy, is described below.
An analysis of the operating conditions of bearing assemblies of various
types shows that, in most cases, the resultant radial force R = (Ry + R^)-^^^ and
the resultant moment M = My + M^)-^^^ absorbed by the bearing act in ona_5Lnd the
same plane. In conformity with this, by laying out the plane of the coordinates
xOz such that it coincides with the plane of action of the external loads ap-
plied to the bearing, we can write
372
(1.9)
As shown by calculations, the load distribution depends little on the angu-
lar arrangement of the set of "balls. Taking this into account, we can assi:ane
that the balls are arranged symmetrically with respect to the plane :xOz. Under
this condition, we have
^0
g
(1-10)
Keeping in mind the equalLties (1«10), we then expand eq.(1.6) in a /336
Maclaurin series in the neighborhood of u = and e^ = e =0. limiting our-
selves to Unear terms we obtain, after easy transformations.
8^ = S + (rf cos ? -f £ sin 3) cos ^,
In the equality (l.ll), we have
^=(52 + cos'^Po)'^'-l
and
S= ta
Vcospo /
(1.11)
(1-12)
(1.13)
The quantities 6 and P are none other than the relative convergence of the
raceways and the angle of contact in the cross section \|r - 90°.
As follows from eq.(1.7),
Sjj^p ^„ ... g+'^isinJ; +72 COS 4/
COS ^^ =
[(s + eisint|; +e2COs4/)2 + (cospo — ^sin4/-|- ucos^f]^^^ '
cos Po — ^ sin ^l; -f ^ cos ^
[(^"-fTi sin ^ +1^ cos 4;)2 -f- (cos Pq— ^sin ^ + u cos ^f]^^^
(1.1^)
Treating the equalLties (1.1!|-) in the same manner as eq.(1.6«), discarding
all nonlinear terms, and making appropriate transformations, we obtain
sinp4,=:sinpri-^^^(M-ecotp)cos6l;
L cos Po J
COS P4,==>C0S P [l + ^^^ (u-lcot p) cos 6].
L cos Po J
(1.15)
373
Having put
wcosp + esinp .
= — ^1
5 •
(1.16)
we can represent eqs#(l.ll) and (1«15) in the form
"8^=^"5(l-|.Xcos^);
cos ^4, = cos p
cos p(
1 I cos(
COS
- — I COS 6 :
.tan2B/xS ^A cosO
(1.17)
(1.18)
The quantity 6 determining the pressure on the tell whose center lies in
the plane xOy can "be e:xpressed in terms of the angles p and Pq:
cosp
(1.19)
The loading zone of the bearing, as is known, can be found from the condi-
tion that b^ = at its "boundaries.
Setting 6^ = in the equality (l.l?), we obtain the following expression
established at the boundaries of the loading zone:
•^;/=cos-(»^).
(1.20)
The relative convergence of the raceways of the races 6^ attains a maxi- /337
mimi 6o at the center of the loading zone, which
is situated in the cross section t - ^q = 0,
if u cos P + e sin p = ^^ ^ 0> ^^ i^ "the cross
section ^ = ^o = 180°, if u cos P + e sin P =
= 6X < (Fig .4.3).
In the case ^^ =0^ < M'o ^ 180° and
^io= "*io^ ^^ ^^ "^^^ ^^^^ *o = 180°, 180° <
< ri ^ 360° and ^(^ = 360° -ri'
It is understandable that eq«(1.20) holds
only if the parameter X exceeds unity in abso-
lute magnitude. If |x| < 1, then the loading
zone will be 360°, i.e., all balls will carry a
load in the bearing; in this case, the quantity 6 is always positive and the
sign of X coincides with the sign of cos ^q» The latter means that^ for bearings
in which all balls are loaded ^ X ^ 1 at ^q =0, and -1 ^ X ^ at ^q = 180°.
Fig .4. 3 loading Zone of
Bearing .
374
Having taken \lr = ^o in the equality (l.l?), we find
^7 means of eq«(l.2), (l.l?), and (l«2l), recalling that 6^
reduce eq*(l»l) to the form
(1.21)
p^^B.vdur
1 + >^C0S4^ \3/2
1 + ^COS 4*1
'0 /
-, we can
(1.22)
Equations (1,19) and (l#2l) show that in the case X = oo, i.e., at a 180°
loading zone 6 = so that p = Pq is independent of the loading level.
Let us introduce into the examination the simi
1
A=
z{\ +Xcos4;o)'
3/2
V(l + Xcos^)^^^cos^-i ^,
(1.23)
where
yfe--l,2, 3.
Here, as in all preceding equalities, the angle -^ can assume only the dis-
crete values that determine the angular position of the loaded balls.
Let us next transform eqs.(1.3) by means of the obtained expression.
After substituting in these equations P^, sin Pj^, and cos P^ by their values
from eqs.(1.22) and (1.18) in conformity with the equalities (1-9) and (1.23)
and taking eq.(l.2l) into account, we obtain
-v^l
^^oSfsin^AX
XI
X&n
COS p
cospo \l + ^cosvl/o sinp / Ji
-^^B.hli^ cos {iJ,X
^Aa
X
COS p
cosPo
ftM?2
Mi
X5n
M
+ X cos ^Q sin p / -^2 .
X
Xon
COS p
cosfo U 4-Xcostj^o
sin? ; Jzl
il'2U)
375
Equations (1»19) and (1.2l) yield the folio-wing expression for the angle p:
a COS fin
COS 3==- rL^ .
l^j^ (1.25)
The equalities (1*2^) and (1.25) constitute relations which, in engineering
calculations, can replace the "exact" equations of static equili"brium of radial
and radial-thrust "ball bearings. As shown "by actual investigations, the error
produced "bj this substitution in the end results usually does not exceed a few
percent •
When changing the number of "balls, the sums (l#23) vary only slightly. This
permits expressing them in terms of the integrals
Jk-~
2;i(l + Xcost;;o)^^^
\ (l+XcOSt!^0COS6)3/2cOS*-l6^6,
(1-26)
which are a function of the product \ cos ijfo* Here, k = 1, 2, 3.
It is easy to demonstrate that, with the usual number of balls, we have
y^;^cos*-i^oy.fe. (1.27)
The values of the integrals \ are given in Table 4*2.
Table ^
4.2
XcostJ/o
h
h
h
w
X cos tpo
h
h
h
0.210
w
1.000
0.000
0.500
1.000
3.33
0.323
0.247
0.612
0.1
0.868
0,065
0.435
0.879
5
0.309
0.242
0.207
0.605
0.2
0.766
0.114
0.385
0.804
10
0.294
0.236
0.203
0.596
0.3
0.685
0,151
0.346
0.757
20
0.286
0.233
0.201
0.59
0.4
0.622
0.180
0.316
0,726
±oo
0,279
0.229
0.199
0.587
0.5
0.570
0,202
0.292
0.705
-20
0.271
0.225
0.197
0.583
0.6
0.528
0.220
0.273
0.690
-10
0.262
0,221
0.194
0.578
0.7
0.494
0.233
0.258
0.67S
-5
0.247
0.212
0.188
0.567
0.8
0.466
0.243
0.246
0.670
-3.33
0.229
0.201
0.181
0.55'6
0.9
0.443
0.250
0.237
0.663
-2.5
0.211
0.189
0.172
0.543
1
0.425
0,255
0.231
0,657
-2
0.192
0.175
0.162
0.528
1.111
0.409
0.257
0.226
0.651
—1.667
0.171
0.159
0.149
0.512
1.25
0.395
0.258
0.223
0.645
-1.429
0.147
0.140
0.133
0.488
1.429
0.380
0.258
0.220
0.639
-1.25
0.120
0.116
0.112
0.459
1.667
0.366
0.256
0.218
0.633
-1.111
0.084
0.083
0.080
0.414
2
0.352
0.254
0.215
0.626
-I
0,000
0.000
0.000
0.000
2.5
0.338
0.251
0.212
0.619
376
Section 2» Calculation of Radi al and Radial- Thrust Ball Bearings Zl22
und_e_r Combined L oads, for Absence of Misalignment
of the Races
1. Pressure on Balls
If the distance between the bearings is large in conparison with the dia-
metral dimensions of the bearings and if all components of the bearing assembly-
have a high rigidity, then, in calculations of the pressures on the rolling
bodies, we can disregard the misalignment of the races under load and take into
account only their displacements in radial and axial directions.
Let us introduce the quantities 6q and X into eq.(l.22) which determines
the pressures on the balls in radial and radial-thrust ball bearings.
Equations {1»2U) and (1.25) which connect these quantities with the external
loads appHed to the bearing, in the absence of misalignment of the races, i.e.,
in the case ?^ = 0, can be represented in the form
zvd
bA \ cos Po 1 + ^ yi /
-: ^oB3/2 COS ?j' ( 1 -L. J^^ilian2 fi _1^ A^ .
(2.1)
cos? = ^^^. (2.2)
1 + X
We will not write out the expression for the moment since, at ^? = 0, it
does not play an independent role and is not used in the calculation.
We will assume, for convenience, that the direction of the z-axis coincides
with the direction of the radial load R. Under this condition, the radial dis-
placement u is positive, and hence the angle ^q ^s equal to zero. This fact is
taken into account both in eqs.(2.l) and {2»2) and in all subsequent relations.
It should be noted that the case ^ = is fundamental in the theory of anti-
friction bearings. Usually, when no special stipulations are made as to design
and characteristics of loading a bearing assembly, this is the case applicable.
Basic investigations (Refs.22, 23, 29, and 42) have been carried out to refine
the calculation of antifriction bearings working under combined loads.
The static load capacity of a bearing is characterized by the magnitude of
maximum pressure on the rolling body.
According to eq.(1.22) the maximum pressure on the ball is
Po-~Bovdl^d\ (2.3)
For protracted static loads on a nonrotating bearing, the maximum bearing
377
stress On AX on the track of the iraier race caused "by this pressure should not
exceed 40,000 kg/cm^. If the static loads acting on a nonrotating "bearing create
greater contact stresses, then noticeable traces of residual deformations, in
the form of depressions made by the halls, will appear on the track.
The indicated permissihle value of Onax is selected from the condition that
the extent of residual deformation (permanent set) of the track is not more /340
than one micron per centimeter of the ball's diameter. In this case, the smooth-
ness of the "bearing rotation is not disturbed and the bearing capacity is not
lessened.
In the relations reqioired to calculate the life e:xpectancy of bearings, the
quantity
(2.4)
appears, where m is the e:xponent of the load in the life expectancy formulas.
By means of the equalities (1.22) and (2.3)* we reduce the expression (2.4)
to the form
^e,= ^^^0. (2.5)
The coefficient w here is equal to
w^=^
~ ^^^— - [ ( 1 -|- X cos % cos ^)3/2'" d'^
V,
(2.6)
We note that the quantity P^q is the constant pressure P^ = const, at which
the probability of fatigue failure of the rotating race under the given service
conditions is the same as for the actual distribution of forces between the
balls. This justifies denoting it as the equivalent pressure on the ball for a
rotating race.
It should be borne in mind that, in some cases, it is inpossible to relate
the quantity P^q to the entire length of the track as is done in eq.(2.4), but
only to the loaded zone |^^ " K •
At m = 3 •33, as is adopted in Soviet practice,
1
w =
jt(i + Xcost;;o)5L 2 \ 8 / (2.7)
. ,. / 137 , 607.0 I 8,4^ '^'^
^'neO ' 120 * 15
378
For practical application of eqs«(2«6) and (2»7) it must be remembered that,
for the selected direction of the z-axLs, the angle iItq is equal to zero* The
angles ^^'^ and ^^^ in eq.(2«7) are taken in radians. The values of the coef-
ficient w found by this formula are given in Table 4«2, together "with the values
of the integrals i^ .
The pressures Pq and P^q at the given external loads R and A can be calcu-
lated in two ways.
The first consists in calculating these quantities by means of eqs.(2.3)
and (2.5)* making use of the values of 60 and \ obtained
from a direct solution of eqs.(2.l) and (2.2).
Since eqs.(2.1) and (2.2) have a coiiplex structure,
it is logical that this procedure encounters great diffi-
culties. These are still large, even when the problem is
solved approximately.
/
The second way, more acceptable for practical use in
determining the pressures Pq and P^q is based on the fol-
lowing considerations:
If the angle of contact of all balls is the same /341
p^ = P = const, then
zvd
b&
■^5o?>o'/2sin?y,;
Fig .4 .4 Resultant
of Force AppHed
to Bearing.
6a
(2.^)
The relations (2.8) differ from eqs.(2.l) in that they
do not contain terms allowing for the variation in angle
of contact as a function of the position of the ball relative to the plane :xOz.
For a 1B0° loading zone, when half of the balls are operative in the bear-
ing, we have X = ±00 so that p = Pq, ji = 0.279, 32 = 0.229 and w = 0.5B7.
For the given case, eqs.(2.8) yield
^ J2
(2.9)
such that
Po - ^ -4.37 ^^
2 C0S^J2 ZCQS\
•* z cos
(2.10)
379
TABUS I
..3
\
a
/'^
\
0*
10"
20°
30°
40°
50*
60°
70°
80°
'-<
\
/342
Values of the coefficient k^
0.02
1.000
0.04
1.000
0.07
1.000
0.11
1.000
0.14
1.000
0.21
1.000
0.35
1.000
0.53
1.000
0.70
1.000
1.00
1.000
Po =
0.889
1.070
1.210
1.308
1.380
1.422
1.428
1,398
0.890
0.966
1.091
1.146
U196
1.212
1.192
1,152
0.880
0.924
1.014
1.078
1.116
1.116
1.094
1.050
0.920
0.880
0.958
1.010
1.034
1.030
1.000
0.950
0.926
0.873
0.933
0.974
0.994
0.984
0.952
0.906
0.936
0.858
0.898
0.930
0.938
0,916
0.880
0.830
0.948
0.858
0.858
0.876
0.872
0.850
0,804
0.746
0.956
0.874
0.836
0.842
0.828
0.798
0.750
0,686
0.961
0,882
0.824
0.822
0.802
0.748
0.718
0.648
0.967
0.893
0.818
0.800
0.774
0.734
0.678
0.602
0.02
1.399
0.04
1.249
0,07
1.184
0.11
1.136
0.14
1,114
0.21
1.086
0.35
1,056
0.53
1.038
0.70
1.028
1.00
1.020
0.02
1.628
0.04
1.453
0.07
1.327
0.11
1.252
0.14
1,217
0.21
1.169
0.35
1.108
0.53
1.073
0.70
1.056
1.00
1.035
Po-12°
1,171
0.839
0-897
0.963
0.999
1.005
0.9921
1.109
0.842
0.875
0.933
0.959
0.956
0.938
1.077
0.847
0.858
0.902
0.918
0.914
0.879
1.046
0.853
0.841
0,871
0.881
0.870
0.831
1,034
-.0.860
0.831
0.859
0.862
0.847
0.809
1.018
0,870
0.816
0.833
0.833
0.809
0.771
1.008
0.889
0.803
0.806
0.799
0.781
0.721
1.002
0.899
0,802
0.789
0.772
0.734
0.688
0.995
0.908
0.810
0.778
0.754
0.715
0.665
0.983
0.912
0.813
0.761
0.734
0.694
0.636
0,937
0.883
0.624
0,777
0.754
0.713
0.655
0.623
0.601
0.567
?o = 18°
1.476
1 ,'001
0.785
0.804
0.813
0.804
0,770
0.718
1.338
0.981
0,818
0.797
0,799
0.789
0.759
0.697
1.238
.0.961
0,786
0.780
0'.782
0.766
0.727
0.672
1.172
0.951
0,781
0,777
0.770
0.749
0.707
0.649
1.140
0.948
0.782
0.771
0.761
0.741
0.696
0.637
1.108
0.941
0.784
0.763
0,749
0.719
0.675
0.612
1.058
0.932
0.789
0.748
0.730
0.694
0.646
0,582
1.027
0.924
0.789
0.739
0.717
0.671
0,618
0.554
1.018
0.926
0.797
0.736
0.707
0.661
0,604
0,536
1.008
0.915
0.806
0.732
0.682
0.647
0,590
0.518
90°
1.308
1.080
0.976
0,874
0,826
0.750
0.667
0.602
0.566
0.526
6.877
0.808
0.752
0.705
0,681
0.636
0.579
0.538
0.514
0.479
0.647
0.625
0.597
0.572
0.560
0.534
0.496
0.466
0.455
0.429
3S0
TABLE 4.3 (contM)
/2^
0.02
0.04
0.07
0.11
0.14
0.21
0,35
0.53
0.70
1.00
0,02
0.04
0.07
0.11
0.14
0.21
0.35
0.53
0.70
1.00
0.02
0.04
0.07
0.11
0.14
0.21
0.35
0.53
0.70
1.00
2.171
1.815
1.629
1.496
1.445
1.343
1.207
1.129
1.098
1.053
10*^
20*»
30°
40°
50°
60°
70°
80°
90*
Po = 26°
1.914
1.815
1.478
0.872
0.692
0.667
0.641
0.594
0.530
1.609
L528
1.321
0.865
0.692
0.665
0.635
0.587
0.523
1.483
1,415
1.218
0.858
0.692
0.661
0.629
0.577
0.514
1.389
1.326
1.150
0.850
0.692
0.656
0.621
0.566
0.504
1.336
1.279
1.119
0.847
0.692
0.655
0.618
0.560
0.500
1.249
1.204
1.069
0.845
0.692
0.650
0.611
0.551
0.492
1.175
1.125
1.011^
0.837
0.693
0.647
0.600
0.544
0.480
1.121
1.078
0.975'
0.828
0.696
0.642
0.593
0.535
0.466
1.087
1.051
0.955
0.826
0.698
0.639
0.589
0.527
0.455
1.051
1.016
0.932
0.819
0.701
0.632
0.580
0.515
0.440
0.02
1.237
0.04
1.125
0.07
1.093
0.11
1.065
0.454
0.447
0.440
0.431
0.424
0.413
0,398
0.383
0.373
0.358
7,051
1.756
1.567
1.425
1.359
1.262
1.156
1.086
1.053
1.006
1.823
1.574
1.403
1.272
1.205
1.144
1.060
0.999
0.967
0.925
Value
1.426
1.242
1,132
1.052
1.011
0.970
0.913
0.870
0.848
0.821
= 36°
0.795
0.777
0.757
0,746
0.742
0.736
0.724
0.714
0.707
0.696
0.580
0,575
0,574
0.578
0.578
0.578
0.578
0.578
0.578J
0.5781
0.507
0.507
0,514
0.514
0.514
0.510
0.510
0.509
0.506
0.502
0,455
0,453
0.453
0.451
0.449
0.449
0.445
0.444
0,440
0.434
0.388
0.388
0,385
0.380
0,379
0.375
0,370
0.364
0.364
0.356
of the
coefficient k
= 0°
1.000
0.995
1.297
1.520
1.710
1.882'
2.030
2.150
2.230
1.000
0,962
1.150
1.350
1.502
1.655
1.758
1.825
1.870
1,000
0.950
1.055
1.212
1.355
1.475
1.540
1.608
'1.680
1.000
0.968
1.005
1.130
1.232
1.325
1,400
1,452
1.477
1.000
0.966
0.985
1.093
1.190
1.275
1,330
1.375
1.395
1.000
0.963
0.952
1.045
1.420
1.175
1,225
1.260
1.275
1.000
0.970
0.937
0,979
1.041
1.082
1.107
1.122
1.133
1.000
0,972
0,934
0,944
0.986
1.015
1.025
1.032
1.030
1.000
0.977
0.934
0-928
0,952
0.973
0.976
'0.972
0.968
1.000
0.982
0.935
0.907
i
0.919
lo = 12
0.926
0.922
0.905
0.896
1.090
1.051
1.034
l.OU
0.929
0.919
0.913
1.039
0.991
0.979
1.190
1.142
1.082
1.301
1,235
1.159
1.384
1.299
1.213
1.438
1.333
1.257
0.9131 0.9631 1.040 l.lOOl 1.146| 1.1811 1.188 1.197
1.480
1.369
1.270
0.310
0.307
0.304
0.299
0.299
0-299
0.291
0.285
0,283
0.275
2.270
1,900
1.710
1,490
1.408
1.283
1.140
1,029
0.966
0.890
1.489
1,377
1.280
381
TABliE 4-3 (cont»d)
0.14
0.21
0.35
0.53
0.70
1.00
0.02
0,04
0.07
0.11
0,14
0.21
0.35
0.53
0.70
1.00
0.02
0.04
0.07
0.11
0,14
0.21
0.35
0.53
0.70
1,00
0.02
0,04
0.07
o.n
0.14
0.21
0.35
0,53
0.70
1,00
IM.
1.053
1.037
1.017
1.010
1.006
1.001
1.336
1.231
1.170
1.125
1.104
1.067
1.041
1.024
1.013
0.999
1.451
1,285
1.215
1.182
1.150
1.096
1.043
1.016
1.005
0.989
10°
1.006
0,999
0.994
0.988
0.982
0.976
20»
0.914
0.916
0.923
0.929
0.931
0.933
30*»
0.951
0.921
0.897
0.885
0.883
0,873
40**
50°
1.017
0,981
0.937
0.911
0.891
0,870
1.071
1,022
0.959
0.927
0.897
0.864
60"
1.111
1.051
0.983
0.931
0.892
0.851
70"
r.i36
1.057
0.986
0.925
0.885
0.834
80"
1.149
1.082
0.991
0.918
0.877
0,826
90"
Po=18'
>
1.260
0.972
0.879
0.944
1.003
1.048
1.070
1.091
. 1.175
0.962
0,873
0.931
0.979
1.008
1.036
1.056
1.115
0.952
0.866
0.915
0.957
0.981
0.999
1.017
1.077
0.941
0.860
0.899
0.933
0.957
0,965
0,973
1,057
0,937
0.856
0,889
0.920
0.942
0.944
0.951
1.033
0.932
Q.856
0.872
0.894
0.906
0.906
0.913
1.011
0.935
0.856
0.856
0.865
0,865
0.865
0.858
0.996
0.934
0.856
0.837
0.846
0.836
0.825
0,810
0.986
0.932
0,856
0.830
0,830
0.818
0,799
0.780
0.974
0,929
0.856
0.819
0.806
0,789
0.770
0.751
i.4ir
1.249
1.195
1.159
1.120
1.071
1.021
0.992
0.971
0.953
Po = 26'
1.227
0,863
0.775
0.778
0.782
0.778
0.775
1.132
0.863
0.773
0.776
0.773
0.772
0.767
1.078
0.859
0.772
0.769
0.764
0.761
0.754
1.034
0.856
0.769
0.762
0.754
0.749J 0.741
1,011
0.854
0.768
0.758
0.749
0.741
0.732
0.980
0.850
0.766
0.755
0.741
0.729
0.719
0,959
0.847
0.764
0.746
0,728
0,707
0.692
0,936
0.841
0,764
0.737
0.714
0.690
0.668
0.923
0.836
0.764
0,732
0.705
0.678
0,652
0.905
0.836
0.764
0.728
0.701
0,658
0.634
Po = 36'
)
1.525
1,480
1,363
1.149
0.778
0,643
0.603
0.566
0,540
1.367
1.333
1,124
1,060
0.769
0.639
0.603
0.566
0.536
1.269
1.233
1.141
0,990
0.756
0.638
0,600
0.564
0.534
1.195
1,157
1.076
0.^42
0.748
0,637
0.599
0.558
0.531
1.157
1.120
1.045
0,922
0.746
0.638
0.599
0,557
0.527
1,098
1,066
0.996
0.888
0.744
0.639
0.597
0,554
0.520
1.035
1.012
0.946
0.858
0.736
0.639
0.592
0,550
0,511
0.993
0.970
0.910
0,832
0.728
0.639
0.589
0.544
0,502
0.970
0.946
0,890
0.815
0,723
0,639
0.585
0.540
0,500
0.945
0.914
0.863
0.798
0.718
0.639
0.582
0,534
0.493
1.152
1.077
0.989
0.917
0.869
0.812
1.102
1.062
1.013
0.965
0,942
0.906
0.856
0.808
0.780
0.737
0.767
0.759
0,747
0,732
0.722
0.705
0.678
0.652
0,634
0.615
0.526
0.525
0.521
0.518
0.514
0.508
0.498
0,491
0.485
0.473
382
Equations (2»10) are known in the theory of antifriction "bearings as "St:.*i-
beck formulas".
For the case 7^ 1«217 tan Po the pressures Pq and Peq can he repre-
sented as ^
Po = 4.37
KqF
^.-2.57
Z COS po
kF
(2.11)
"^ Z COS po
where F = (R^ + A^)"^^ is the resultant load on the hearing (Fig. 4*4) •
This same form of notation can also he retained when taking account of the
variability of the angle P^ •
After substituting the values of Pq and P^^ from the equalities (2»3) and
(2.5) into eq.(2.11), we obtain
/5y5^'^cos?o
4.37
Kn
p
w
0.587 ^
(2-12)
The coefficients ko and k are unique reduction coefficients referred to
the resultant load F. It is essential that they can be found from eqs.(2.l) and
(2*2) by different indirect methods which preclude the need for direct solution
of these equations.
The values of the coefficients ko and k for bearings with initial angles
of contact Pq = 0> 1^* ^f ^^ 36°, obtained from eqs.(2.l) and (2.2) by the
graphoanalybic method (Ref .30), are given in Table k*3»
Introduction of the tabulated reduction coefficients ko and k greatly ^345
facilitates finding the pressures Pq and Pgq, permitting the use, for this
purpose, of rather siirple and convenient formulas [eqs.(2«ll)] .
The coefficients ko and k are given in Table 4»3 as a function of the
quantity , which characterizes the level of the load received by the
bearing, and of the angle a = tan •*- ., which deter*mines the direction of the
resultant F--". ^
F R A
"^''' In calculating the quantities and also as well as ;
ZVd^a ZVd^a 2Vd'
the diameter d^a is always e:xpressed in millimeters.
ba
383
2. Reduced Loads
Let us denote by Q the radial force which, in comlDination with the axial
force A = 1.21? tan PqQ at a constant contact angle between balls and races
P)lr = Po = const, creates the same equivalent pressure P^^ as the actual combina-
txon of external loads applied to the bearing •
The force Q is connnonly called the "reduced dynanac load".
Along with the concept of reduced dynainic load, the concept of "reduced
static load" is widely used in the theory of antifriction bearings. By reduced
static load we mean the radial force Qq which, under the indicated conditions,
exerts a maximim pressure on the ball Pq equal to the actual pressure.
The replacement of actual loads by reduced loads, determined as indicated
above, permits using data from catalogs and handbooks of radially loaded bear-
ings, when calculating bearings operating under combined loads.
A conparison of the equalities (2.10) and (2.11) shows that
Qo==K,F. 1 (2.13)
In other countries, and recently also in domestic use, a formula of the
following type is often used to determine the reduced loads:
Q = xR+yA.
Different sources give different values of the reduction coefficients x and
y, so that the reduced loads calculated for one and the same case may differ
substantially.
Since all calculation methods for radial and radial-thrust ball bearings
under combined loads, used in practice, are based on the same initial equations
[eqs.(l.l) - (1.7)] and basically differ only by the assuirptions used for sim-
plifying their solution, one of the principal criteria of the quality of one or
another calculation method is the closeness of the reduced loads calculated on
its basis to the "exact" value of these loads obtained from the indicated equa-
tions.
Figure 4.5 gives a conparison of the reduced loads determined by means of
the coefficients of Table 4*3 with the reduced loads found as the result of the
"exact" solution of eqs.(l.l) to (I.7). The same diagram shows the reduced / 346
loads calculated by the method of the International Standards Organization (ISO^O
recently adopted in other countries and calculated by the method of M.P.Belyan-
chikov (Ref .24) which is now being recommended for the calculation of general-
purpose bearings.
""" Draft of recommendations for calculating dynamic load-carrying capacity of
ball and roller bearings, ISO, No. 278, I96O.
3^k
Fig.4-5 Comparison of Various Methods
for Calculating Reduced Loads on a
Bearing.
As we see from rag.4«5, the re-
duced loads obtained by using the
data of Table 4-3 are closest to
their "exact" values.
Use of the ISO method, under
certain conditions will overestimate
the reduced loads by 20 - 30^ which
is naturally impermissible for bear-
ing assemblies of aircraft com^
ponents •
Sufficiently accurate ^values of
the reduced loads are obtained with
the method developed by M.P.Belyan-
chikov for calculating radial-thrust
ball bearings with contact angles of
00 ^ 26°. However, at smaller con-
tact angles, the accuracy of the
method decreases steeply. For in-
stance^ in the case of contact
angles of Po = 12 - 18°, the error
in the reduced load may go as high
as k-Ofc* For contact angles less
than 12°, this method is generally
unacceptable.
3* Statistical Theory of Dynamic /347
Load-Carrvine: Capacity
In calculations of life ex-
pectancy we generally use the prin-
ciples of the statistical theory of
fatigue of metals, which assumes that failure of the material under the effect
of alternating loads is a random process of accimaulation of fatigue cracks having
various probabilistic characteristics. Such an approach to the problem of life
expectancy is highly useful for any machine coirponent operating under alternat-
ing stresses, including anbifrictions bearings which fail as a consequence of
fatigue chipping of the tracks or rolling body.
Statistical representations, underlying modern methods of determining the
service life of antifriction bearings, were developed mainly by Weibull (Ref .43)
and Lundberg and Palmgren (Ref.44)» Investigations by Harris (Ref.45) and
others were devoted to the development of these representations for small proba-
bilities of failure.
The basic principles of the statistical theory of the dynamic load capacity
of roller bearings can be formulated in the following manner:
^"t qt)o t)e the probability that the bearing, rotating at an rpm of n, works
h hours without signs of fatigue.
385
On the basis of the theorem of mathematical statistics for the product of
independent events, disregarding the probaTbilitj of failure of the roller by
virtue of its smallness in conparison -with the probability of failure of the
tracks^ we can write
where q^ot ^^ %t ^^® '^^^ corresponding probabilities characterizing the reli-
ability of the rotating and stationary races .
Taking into account the characteristics of the state of stress under the
effect of contact loads and the character of the primary fatigue microcracks
formed in roller bearings, Lundberg and Palmgren introduced the following dis-
tribution determining the probability F^ of the appearance of traces of fatigue
on a portion of the track of length AL after N rollers loaded by a constant
force P have rolled along it:
F,^l^exp^^H,^^^^-^v\ (2.15)
Here,
Hi = coefficient depending upon material properties, surface finish, and
precision of manufacture;
To = maximum tangential stress acting in areas parallel to the surface of
the area of contact strain;
Zq = depth at which this stress arises;
AV = stressed volume.
At ms = 0, which might occur when the probability of failure introduced by
each element of vol-ume does not depend vpon its location relative to the siu?face,
the distribution (2.15) changes to the customary Weibull distribution.
The stress Tq and the depth Zq can be expressed, respectively, by the maxi-
mum bearing stress Oq at the center of the area of contact strain and the semi-
minor axis b of this area:
The stressed volimie AV, in first approximation, can be taken as equal to /^3U8
Al/-2azoAL, (2*1?)
where a is the semimajor axis of the area of contact strain.
As follows from the theory of contact stresses and strains, for radial and
radial-thrust ball bearings.
386
4100 / . , 2
i^v
a = 0,0108M./ 4-
Tl
i^-=0.0108v
f-
T1
2
T 1
'^^4;
^1 ^^^^•
(2.1S)
\ — +1 /
In the equalities (2.18) the following notations are adopted:
The coefficients H^, a^, andcv^, strictly speaking, are not constants; how-
ever, for all practical purposes this can be disregarded since the Umits within
which their values vary (depending on the ratio b/^) ^^® quite negligible.
It is easy to prove that
AL = ^^^^cos^ (^-Lqzi^^,^,
(2.19)
where At is the central angle corresponding to the examined portion of the track-
By means of the equalities (2.16) - (2.19) we can reduce eq.(2.l5) to the
form
FA=l~exp I — 7/2 cos
1 2 '^ub 1^0
., P'^^N^ A6
^Aa
2k
(2.20)
The number of balls contacting each portion of the track during h hours of
work of the bearing will be
N=^30znh{\±r]). (2.2l)
Substituting this value into eq.(2.20), we finally obtain
F^=l — exp
■M^z^ COS %(nhy
P""^ Avb
^£ 2n
(2.22)
The e:xponents m and c are e^^ressed in terms of the e^tponents m^, mg, and I
in the following manner:
3B7
m-
SI
2mi-^m2 — 5 *
/Mi+2 — m2
(2.23)
According to the data of foreign "bearing manufacturers, which are gen- /349
eralized in the recommendations of the ISO, m = 3 and c = 1.8 (at d^a < 25 rm) »
The portion of the track of the stationary race located at the azimuth ilr
when the tialls roll along it, is loaded each time "by the same force P^ . Setting,
in conformity with this, P = P^ in eq.(2.22), we obtain for this portxon
f^ = l — exp
■""^^'^-^^o^'^^^y-^^l
(2.24)
It follows from eq.(2.24) that the probability qg^ , characterizing the re-
liability of the stationary race as a whole, is equal to
where
<7,f =n(l--pA;) = exp
^■f*" 2rt
^m: '
- M,zi COS % {nhy ^-^
(2.25)
\ Pfd'^^
\ml
During a sufficiently long time interval, each element of the track of the
rotating race will contact the balls at practically all azimuths •
Accounting for this fact and considering the hypothesis of linear sianmation
of damageabiHty to be valid, eq.(2.22) will yield for each portion of the track
of the rotating race
/^A=l — exp
^H^zizo^^,{nhY^^
AO
d'i^ 2^
(2.26)
Here, ^^~^ \ ^'"^'M is the same equivalent pressure discussed in
Subsection 1.
The probability q^ot characterizing the reliability of the entire rotating
race in conformity with eq.(2#26) will then be
r p'"^ 1
(2.27)
388
The coefficient H3, figuring in the above equalities, can be represented as
the product of a certain constant H4 and the quantity
//,
2mi4-ma— 2
4 h^
1=1
(2.28)
which is a function of T| and 0, i.e., parameters characterizing the interior
geometry of the bearing. '
Let us assume that the inner race rotates. For this, eqs.(2.14), (2.25),
and (2.27) will yield
-//.
. ml
«,.+(&-) «..,
z^ COS '^Q (n/i)
I lis^
(2.29)
^mlc *
The indices "in" and "out" (eq.o = equivalent, outer) as well as the /350
upper and lower signs in eqs.(2.18), (2. 19), (2.2l), and (2.28), pertain, re-
spectively, to the inner and outer races of the bearing.
At Tl = 0.2, e = 0.52 and X = 00, let
tiu
P.^.^\^^
xx)
^Oirt^^S-
Furthermore, let us introduce the quantities Cq and f^^ over the formulas
Co=/7'VL^
(2.30)
where
2.57
1 1 nm/
In ■
and
H^H^ 0.9.
/' =
H.
ml
f.-=
log
1
9be
In —
L O.9J
(2.31)
389
Using eqs.(2.30) and (2.31), we transform eq.(2.29) such that
0.39z cos %Pe^{nhr =0^ (cos %)\ '"V/;
1 A J_
(2-32)
Having expressed here IP^^ in terms of the equivalent dynamic load Q, we
obtain
where
1 1
-Cn(C0s3n)^' ""'l
(2.33)
C^Co(cospo)
In like manner, we can examine the case where the outer race rotates.
Combining the formulas of life expectancy for rotation of the inner and
outer races and introducing the coefficients k^ and kt which take into accoimt
the effect of the type of load and tenperatiire regime of the bearing on the load-
carrying capacity, we fimlly have
fc^K^K.Qinh)'^ -C//
1 __i^
(2.34)
Here k^ = 1 if the inner race rotates, and
Ak =
ml
(2.35)
if the outer race rotates.
In the specific numerical calculations of radial and radial-thrust ball /351
bearings on the basis of tabulated coefficients ko and k, the values of the kine-
TABLE 4.4
■n
0.05
0.10
0.20
0.30 0.40
/"
0.77
0.92
1
0.930.81
""^"^
-~^
^0
■ -
^i
■0J37
10
0,8
0.5 OJ 3J 0^8 0.9 w
Pig .4 •6 Kinematic Coefficient kj^.
390
matic coefficient k^ can be detennined approximately as a function of the quan-
k
tity w = 0.587 ^j from the graph in Fig.4«6.
In other countries, calculations of the coefficients of utilization Go and
C usually take the coefficient f ' as equal to I50 - 200.
Calculations show that, at a given 9, the coefficient f depends inainly
on Tl* At m = 3^ C = 1.8, and I = 1.11 which corresponds to the ISO recoiranenda-
tions, this coefficient has the values indicated in Table 4»4«
For general-p-urpose bearings, the Hfe expectancy hxo at which the proba-
bility of failure is equal to 10^, is considered to be the rated life.
Since we have at q = 0.9 a value of fq = 1, it follows that
1
^b^t^KQ(nh,o) ^ =C, (2.36)
Conparing the equalities (2.34) and (2*36), we find
-tr^"- (2.37)
It follows from eqs. (2.37) and (2.3I) that the average life expectancy of
roller bearings is determined from the eiJ^ression
< r
hi)
where T is the gamma function of the arg-ument ( 1 + ^^ •
The ratio of the median life expectancy h^of corresponding to the reliabil-
ity q = 0.5, to the rated life h^o will then be
h
^^{ -^ 1 . (2.39)
Equations (2.38) and (2-39) show that the main parameters characterizing
the dispersion of the life expectancy of roller bearings is the exponent i .
hcQ
In most cases, the ratio varies within limits from 4*08 to 5. At
391
-^^^2« = 4.08, t = 1.34 and -i^^^ = 4-95* At -^^^^ = 5, ^ = 1.17 and ^^^^ = 6.5-
As experimental investigations indicate, these relationships satisfactorily
descrilbe the dispersion of life expectancy at q^^^ ^ 0.9» Noticeatile deviations
are observed in the region of small prolDabilities of failure. These devia- /352
tions can be taken into account if, for this region, eq.(2.37) is replaced ty
the following:
^10-
(2.40)
Here, ho is some threshold of life expectancy, prior to which the probability of
failiire is equal to zero.
Since we have fq ^ [10(1 - q^e)^^^^ at q^^ < 0-9, eq.(2.40) can finally be
T(«*itten in the form
According to Harris' data (Ref .45), the ratio for ball bearings is
h.
hio
?^ 0.045. This means that, to ensure 100^ reliability, a rated life margin of
the order of 22 is required which corresponds to a load margin of 2.8.
The basic principles of the static theoiy of dynamic load capacity of radial
and radial-thrust ball bearings have been presented above. The corresponding
static theories of dynamic load capacity can be developed in a similar fashion
for bearings of other types.
The main results of static representations of the life expectancy of roller
bearings have been applied both in foreign and domestic practice. However, it
should be borne in mind here that some of the fundamental relations used in pre-
paring our own catalogs and manuals have a form differing from that in other
countries. For exairple, our coefficients of utilization Co and C are not calcu-
lated from eqs.(2.30} and (2.33) t»ut are taken as equal to
^^=^^1^^^:^/ ' (2.42)
C-Cocospo. >
For general-piirpose bearings, f = 65. We recall that, in Soviet practice,
the exponent m is considered as equal to 3*33.
The life expectancy of two-row bearings as well as of roller bearings conr-
sisting of several identical bearings which can be regarded as one multirow bear-
ing, is determined by the expression
392
0.39 K^^KfZ COS %fl^. {nhio)'"=C. (2-43)
The equivalent pressure entering here is
e,..
IKV'*)"
Ly-i
(2.-W-)
where P^q and k^ are the equivalent load and kLnematic coefficient for the
j-th bearings •
Equations (2*43) and (2*44) follow directly from the relations given above
for individual bearings .
If all bearings are loaded identically, then
p —f^ifj- p (2«45)
As shown by an analysis of the values of the coefficients f '' in Table 4*4 v/353
the performance of roller bearings largely depends ijpon T]. In eqs.(2»42) this
itrportant fact is not taken into account, which is their essential shortcoming.
As is known, for high-precision aircraft bearings manufactured from particu-
larly high-grade metal, the coefficients of utilization have much larger values
than those obtained from eq.(2.42) for f = 65*
Therefore, using the data of machinery catalogs and handbooks for calcula-
tions of aircraft structures, it can be e:xpected that, in reality, the rated
life hio will not correspond to the 10^ probability of failiore but will be ap-
preciably smaller in value. With this approach to a determination of the service
life of bearing assemblies of aircraft conponents, this life expectancy is often
identified with the required lifetime. In practice, this is achieved by replac-
ing ^10 in eqs.(2«36) and (2.43) "by h, tinderstanding by h the life expectancy at
which the level of reliability of bearings for aircraft conponents is ensiored.
4. Effect of_ Axial Load on Bearing Performance
Let us discuss the manner in which an axial load affects the performance of
radial and radial-thrust ball bearings.
Figures 4-7 and 4*8 show typical graphs of the relation — -^ = f( ) for
= const, plotted from data of calculations performed in compiling the
zvd^
tables of the coefficients k© and k.
393
As "we see from these graphs, for each load level there is a range of values
of -^ in which -^ < 1. Its Iboundaries are given in Table 4*5 •
R R
TABLE 4-5
Values of A/H at Different Contact Angles ^^
deg.
12
18
26
36
0.02
0-0.15
0.26-0.38
0.39-0.66
0.56-0.82
0.84-1.20
0.11
0-0.28
0.25—0.45
0.37—0.60
0.55-0.84
0.76-1.21
0,35
0-0.37
0,22-0.47
0.35—0.63
0.47-0.85
0.38-1.21
1.00
0-0.43
0.04-0.49
0.00-0.65
0.00—0.86
0.00-1.22
At values of — - as indicated in TalDle 4*5, the axial load not only will
R
not reduce the load capacity of the "bearing "but even increase it somewhat. It
is true that this increase is insignificant, since the possible decrease of the
reduced dynamic load is several percent •
In radial-thriost "ball bearings, the "balls are acted xpon by Coriolis forces
which tend to make them rotate about axes perpendicular to the contact surfaces,
li^iction forces arising at the points of contact with the races prevent such
"spinning" of the balls. If there is an unloaded zone in the bearing, then /355
this zone contains no friction forces that would prevent "spinning" of the balls,
and the balls begin to sUde relative to the raceways of the races; at high
rates of rotation, this will lead to overheating and rapid wear of the bearings.
It is logical that, in designing highr-speed bearing assemblies with radial-thrust
ball bearings, it is always necessary to have all balls share the load. In
practice, this is achieved either by installing the bearings at suitable contact
angles or with some a-uxiliary ajcLal load produced by preloading.
The magnitude of the loading zone depends on the correlation between axial
and radial loads applied to the bearing. The greater the ratio A/R, the larger
this zone. As indicated earlier, at ^o = the loading zone is 360° if :^ \ ^ 1.
The value \ = corresponds to the case of axial loading of the bearing in which
the pressures on the balls are identical. The value X = 1 determines the mini-
mum magnitude of the ratio A/R at which all balls are loaded. This, in particu-
lar, follows from eq.(1.22) which shews that, in the case to = ^^ ^ = 1^ "t-^®
force absorbed by the ball located at the azimuth ^i^o = ISO° vanishes.
Taking into account that, at X = 1, we have j^ = 0.425, js = 0.225, and
J3 = 0.231, we find from eqs.(2.l) and (2.2) that
394
Fig»4»7 Graphs of the
Relation -^ = H—^ at
R ^ R /
Certain Constant Values
R
of
and Po =
zvdl
n^t
az OM OS 0,8 1,0 1.1
u i
Fig .4.8 Graphs of the Relation -— = ^("o") ^*^ Certain
R
Constant Values of
2vd|a
R
and Po = 36°.
Fig. 4. 9 Dependence of the Ratio
( ^h^ on Load Level.
\ R A = i
Fig .4*10 Dependence of the Ratio
(" J2^^ on Load Level.
\ R A-i
395
'■k + h+YJ
sin2po + 8o+-r 1—0,6-
-Ji =1.666-^^ *^ ^-^ ^+^°_, ., , ,.,
^(X_l) cospo _ 62 ' (2.46)
sin2po+6o + — — _
1+0.905 — -%r-
cos2Po 2+ Bo
where the radial load R is determined by the expression
-j3/2 ( sin2po+-5o+-^ - I
-V=0-515ocospo—^l 1 + 0.905 ^^ -^ / ,
^v4, '^''2+5o\ ^ cosspo 2+80/ (2.47)
while the ratio is
iilr '■''"' ^-° ir- • ^2.48)
1+0.905— ^-^^ ^~
2+5o cos2po
As we see from the equalities (2.46), (2.47), and (2.48), the values of
( ^^ and ( — ^] depend on the initial contact angle as well as on the level
of the load received ty the "bearing.
For the most frequently encountered initial contact angles 0o ~ 0, 12, 18,
26, and 36°, we plotted in RLgs.4.9 and 4.10 on the basis of eqs.(2.46), (2.47),
and (2.48) curves determining the values ( ) and { -—^^ as a function of
V R /x=i V R A = i
zvd^
ba
For small loads.
With an increase in load, the quantities [ ^ and C*-— ^ will also in- /356
crease. For instance, for the angle Pq ~ -^° ^"t = 1, we have
zvd^a
As shown above, in the case of a constant contact angle between balls and
396
races for a loading zone of 180^ and a parameter X = ±00, the ratio -A. =
- 1.217 tan po* R
The values of the ratio
with consideration of the variation in conr-
V R yx=oo
tact angle as a function of the position of the "ball relative to the plane xOz
can be found from the curves in Pig.4.H» Figure 4-12 shows curves by means of
which the corresponding values of the ratio (— ^^ can be determined.
{/?Lo
0.8
0.6
OA
0,1
u_
'
1
1
-.'J..7/r''
t^'H-^-v^
1 1 60
1
1 —
; 1 '/r
1
1 1
1
. . .
.
-po-n°
0,2
OA 0,6
0,8
zvd
bd.
Fig .4*11 Dependence of the Ratio
(^)
on the load Level.
Q 0,1 0.1 0.3 O.U 0,5 0.6 0.7 0.8
Fig .4*12 Dependence of the Ratio
Q
\=oo
a)
on the Load Level.
\ = oo
The graphs in Figs.4»ll and 4*12 are constructed by means of the formulas
[rIx^^ ° 1 + 0.6
.868un2PoSo '
(~~r) =0,229^0 S"f COS ?o(l + 0M8im^%I,);
V R A^.
1 4-0.868A&n2pQ5Q
(2.49)
which follow from eqs.(2.l) and from the equalities (2.12) and (2*13).
These data permit estimating the effect of the axial load on the load ca-
pacity of radial and radial-thrust ball bearings. With their aid, one can estab-
lish the optimal axial preloading with which the bearings should be mounted in
the assembly and select the most rational values of the initial contact, angle Pq
for different combinations of radial and axial loads.
5» A pproxiaiate Solutions of Equations (2.1) and (2.2)
It should be recalled that the angle Pq is determined by the radial clear-
ance 2A present in the bearing after fitting to the shaft and in the housing at
an established operating temperatiore of the conponent, and also by the actual /357
397
distance g "between the centers Oq and 0^ of the cross sections of the raceways
of the races [see Fig.4«2 and eq#(1.7)].
Because of the effect of shaft-fitting tolerances, nonuniforinity of heating
of individual elements of the assemhly, and possible difference in the values
of the coefficients of linear expansion of the shaft and housing, the clearance
2 A may differ substantially from the initial radial clearance in a self-contained
bearing. Deviations of the radii of the raceways and ball diameter may have a
noticeable effect on the magnitude of the distance g» In this connection, when
calculating highly loaded radial and radial-thrust ball bearings the angle Pq
cannot always be replaced by the rated initial contact angle indicated in a
catalog. This fact must not be disregarded in designing vital bearing assem-
blies of helicopter units and of other aircraft.
The values of the coefficient ko and k for radial and radial-thrust ball
bearings with initial contact angles Pq differing from standard can be obtained
by inteipolation of the data presented in Table 4»3» At the same time, a number
of cases exist in which it is more convenient not to resort to this method but
to solve the problems in the calculation of such bearings by a direct determina-
tion of the quantities Sq ^^^ ^ from eqs.(2.l) and (2.2), using the following
approximate methods.
If -fL > (— ^^ , i.e"-., if all bearings share the load, the integrals can
R V R A=i'
be found from the expressions
yj(l+X)3/2=l+-|x2;
y3(l+X)3.=.-L(i+A>,).
(2.50)
The right-hand sides of the equalities (2.50) represent the first terms of
power series in which the products jic(l + X)^^ have been expanded for ^q =
and ^ X :^ 1. In view of the rapid convergence of these series in the indi-
cated region, the terms containing the parameter X in a power higher than the
third are discarded here.
Solving eqs.(2.l) and (2.2) with consideration of the equalities (2.50),
successive approximations will yield the working formulas
!:(iMT.Ln ^-^^^^V (^-^^^
,1/2 11/3 \ ^+^2 j-
1 \^^^P
398
X=Z),
where
^ + 16-^^
Bozvdi^
(2.52)
The coefficients D^, Dg, D3 are correspondingly equal to
/ J V 1/2 p/3 '
D.=
3 ' A
f 2
cosPo(^l-h —
2 sin2po + 20i O
cos2po
D,^{X-^^^D\)\X-\DI.
32
-Di
^^T?^^
^1
1 +A
Z2^
(2.53)
If — ^ s ( ~^/ * "^^^ formula for the iriaxiiniam pressxare on the loalls can he
represented in the form
^0 — ^0 , •
(2.54)
It follows from eqs*(2.3) and (2.51) that the coefficient ko^^ in eq.(2.54)
can "be equated to
3/2
-^^^lh(nk)"
«(^) =
32 ^
A \>/2
(2.55)
1/2
It is obvious that, in the exajnined case, the reduced loads Qq and Q can be
expressed in the following manner:
(2-56)
399
A comparison of the equalities (2»56) and (2.13) yields, for the given case,
/<:o = 0.229^(^) cos PoSina; ]
(2.57)
K = 0.390'ffi;/i:(^) cos po s in a.
If the radial load is R = then, in conformity" with the equalities (2*53),
we have Dg = and D3 = 1« SulDstituting these values into eq.(2.55), we obtain
the follovjdng for the case of purely axial loading:
1 +
^0 — "^00
sin2 po -f. 2
_M..41/
M,41/
1/2
(2.58)
Figure 4-13 gives graphs of the dependence koo^ = k^o^ (A), obtained from
the initial equations of static equilibrium of radial and radial-thrust ball /359
bearings without sijrplifying assumptions. There, the sign "xn denotes the values
of the coefficient ir^^^ calculated by eq.(2.58), furnishing graphic proof of its
conpletely satisfactory accioracy.
The described method of determining the quantities 6q and X can be used
only when there are no unloaded balls in the bearing.
'Off
5
4
3
2
/
1
—
-
\
Bn =
n
\^
/
-=:
^
^
^^^X
ou
— X-
1
—K
_
Now let the loading zone be less than 360° -
For a loading zone less than 360°, the quan-
tity \ can vary within limits from 00 to 1 and
from -00 to some negative value X-x- corresponding
to the case where the bearing absorbs a purely
radial load. In the absence of an axial load,
the center of the contact area '»slides^' to the
middle of the raceway and the angle P vanishes.
If, in eq.(l.l9), the angle p is equated to
zero and at radial loading of the bearing, we
obviously have
ao/
a 02 a03 O.Qk R
Fig .4.13 Graphs of the De-
pendence k^^^ = k^^^
(A).
Let us introduce the notations
(2.59)
(1 + X)y|-
r2/3
^^2—1 , ^ ' .5/3'
E —A.
72
(2.60)
The values of E as a function of \ are given in Table 4 '6.
At 6 = cos Po - 1 ^.nd P = 0, the second equation of the system (2.1) can be
400
represented as
d2/3 „ cosPo — 1
(2.61)
where
Bozvd'^
¥e recall that, in conforinit7 with the^accegted direction of the z-axis, /360
the angle ^o " ^ and thus X cos to ~ ^ ^-^^ ^o ~6(1 + X)«
TABI^E 4.6
1
X
El
1.244
^2
1.125
^3
1.666
^4
0.600
1
X
—0.1
^1
-0.304
^2
2.670
^3
1.185
B,
1
—8,441
0.9
1.173
1.149
1.594
0.697
-0.2
—0.704
3.126
1.164
—4.296
0.8
1.097
1.187
1.532
0.816
-0.3
-1.248
3.146
1,139
-2.927
0.7
1.017
1.242
1.476
0.968
—0.4
-2.021
4.597
1.115
—2,243
0.6
0.930
1.317
1.430
1.166
-0,5
—3.191
5,899
1.093
-1.829
0.5
0.831
1.409
1.386
1.443
-0.6
-5.111
7.998
1.075
-1,550
0.4
0.717
1.522
1.346
1.857
-0.7
-8.663
11.74
1.051
—1.360
0.3
0.586
1.662
1.309
2.546
-0.8
-16.91
20.23
1.032
-1.210
0.2
0.429
1.836
1,277
3.915
-0.9
47.49
52.48
1.015
—1.094
0.1
0,238
2.055
1.247
8.021
-1
CO
oo
1
—1
2.321
1.217
oo
It follows from the equality (2.62) that X.- should satisfy the condition
A(U-
cospo— 1
^/3
(2.62)
To solve eqs.(2.l) and (2.2) in the case of -^— < f \ , we proceed in
the following manner; Assuming the parameter \ as known, iteration of the second
equation of the system (2-1) will furnish the quantity 6 = 2 — ^ Replacing
1 + X
the trigonometric functions of the angle P "by the corresponding values from
eq.(2«2), we take the following as an approximate value of 6:
n \2/3
' \cos Po '
(2.63)
401
where
2/3
V 2/3 / — .4/3 •
^ .2/3 sin2Po + 2Hi(-^j +£U-V)
« \ Vcos po ^ * 'cos Po '
1 + £, (^ . "" '^-Po^ "^ '^J9s,Po ^ (2.64)
cos2E
_ It should "be borne in mind that, since eq«(2»63) is approximate, the value
of 6 determined from this e:xpression for X = X.- will differ somewhat from the
value corresponding to eq.(2«59)»
riarthermore, from eqs#(2«l) we have
R cospo sin2po4-2S + S 2 E^ 5_ \ 1 + 5 / (2.65)
C0s2po ' ^1 ' 1 + 5"
Prescri'bing X^ we then use eqs.(2«63) and (2.65) for plotting the graph of
the dependence = F(\) (Pig.4«14)» From this graph, knowing the ratio — -,
R R
we find the value of \ which constitutes an approximate solution of eqs.(2.l)
and (2.2)_. Using the obtained value of \, we calculate "by eq.(2.63) the actual
value of 6 and then find 60 from it.
As shown by numerical calculations, the accuracy of eqs.(2.63) and (2.65),
just as of eqs.(2.5l) and (2.52), is conpletely siofficient for engineering ap-
plications. The deviations of the values of 6q and \ calculated by the indi-
cated formulas from the corresponding "exact" values determined by eqs.(2.l) and
(2.2) for initial contact angles of Pq < 45° are usually no more than 3 - l^%»
On the basis of eq,(2.63), we can write
/'o = ^oV<(l + Xf^G.£f\-^. (2.66)
1 — R
Recalling that Ex = and R = , we find the followinf^ /361
, , ,, (l-X)j|/= Bozvd-,. ^"^
for the case -^ < { -^
R \ R A=
1
^°-'^^^'T^- (2.67)
Here,
/C^'^'^^L. (2.68)
/2
402
Fig .4 •14 Auxiliary Graph
for Approximate Solution
of Eqs#(2*l) for a Load-
ing Zone less than 360° .
A coarparison of the equalities (2»ll) and
(2.67) readily shows that the coefficient ko in
this case can be e:xpressed as
A:o = 0.229/cWcosa.
Accordingly,
k = 0.390wkI^^^ cosa.
6. Relative Displacements of Races
For certain ultra-precision high-speed bearing
assemblies, a proper determination of the relative
displacements of the bearing races under load is
of in^Dortance . When combined loads are absorbed by radial and radial-thrust
ball bearings, this problem is solved in the follovdng manner:
Equations (1.13) and (1.16), in the absence of mutual misalignment of the
^— = ) , indicate that
races ( e = ?^ '
s=t0?P cos %;]
u = -
(2.69)
cosp
On the basis of eq.(l.l9), '^e find
5 = (sin2po + 28 + 52)i^';
- X5 (1 + b)
(2-70)
Let
Under this condition, disregarding the quantity 6'
R \ R A=i
owing to its smallness in the equalities (2*42), the following e^^pressions are
obtained from eqs.(2»5l) and (2»52) for the relative displacements i and u:
s^
sin2Po+2M^.
O2
T\2y3
D.
16
■Dt
1 +
\l/2
2/3
I.4ID3J
A \^/2
1/3
1/2
32 ^
■^\2j3
1 +
IAW3
1/2'
2/3
sin2Po+2
1/211/3
I.4IO3
n
(2.71)
403
It is easy to define the variation in the displacements s and u "with any /g62
change in the ratios of radial and axial loads on the exanple of a 3620? bear-
ing, for which the curves u =
\-
R
zvd^
ba
-)
and s = s'
R
^ 2Vdta
are plotted by means
of eqs.(2.7l) in Fig .4.15 ^ot a constant value of the axial load f — =
V zvd^a
= 0.53)* For conparison, the sign "x" indicates the exact values of displace-
ments s and u calculated by eqs.(1.3)^ (1»6), and (1.?) for the cases X = and
?\. = 1, which determine the limits of applicability of eqs.(2.7l).
As follows from the presented data, the equation of moments does not enter
into the system of equations by means of which we investigate the distribution
of the load in radial and radial-thrust
ball bearings operating without misalign-
ment of the races. Therefore, the assunp-
tion of the effect of the radial force
and moment in one plane, which was used
in deriving eqs.(l.24), introduces no
additional limitations that would narrow
the range of applicability of the afore-
mentioned method of calculating such
bearings .
s
?r-
s ■
[=1
OA
— v<-
[
'">
^
^
A = 1
0.3
—
n n
<-l
10
0.8
0.6
OM
02
0.1 O.l 0.3 0> 0.5 O.B 0.1
2Vd
Fig -4-15
and
Curves of u
u'v
ba
R
s = s(
^ zvdta
Value of the
zvd?
"ba
■ ) at a Constant
Axial Load.
Until now, we had assimied the radial
and axial loads acting on the bearing as
given.
The radial loads on bearings are
found from the equations of equilibrium
of the shaft to which they are fitted.
At large distances between the indi-
vidual supports, a determination of such loads is not difficult since, in this
case, they depend little on the moments absorbed by the bearings so that these
can be disregarded in the calculation.
Often, considerable difficulties are encountered in calculating axial loads.
Strictly speakirig, the axial load can be considered as known only in the case in
which the bearing in question absorbs the entire axial force applied to the
shaft, as takes place in bearing assemblies with one bearing fixed in an axial
direction.
Of course, if the equations of equilibriiM of the shaft are not sufficient
for finding the loads acting on its supports, it is irrpossible to make a separate
calculation of bearings mounted on separate supports. In such cases, the pres-
sures on the balls can be determined only by solving the eqtiations of equilibrium
of the shaft simultaneously with the equations of static equilibrium of all bear-
ings fitted to this shaft.
404
Section 3» Certain Prolpleins in Calculatir^g Radial-Thrust Ball
Bearings with Consideration of Misalignment of
their Races under Load
1. Basic Rej^tionships
In a nimiber of helicopter units, narrowly spaced radial-thrust ball "bear-
ings absorb combined loads in which the moment plays, an appreciable if not the 7363^
main role. It is understandable that, in determining the parameters character-
izing the performance of such bearings, it is inper-
missible to disregard the misalignment of the races as
had been done in the preceding Section; this greatly
coiiplicates their calcuH^tion.
The absence of reliable methods for calculating
radial-thrust ball bearings receiving appreciable mo-
ments at close spacing of the sijpports interferes with
the design- of numerous bearing assemblies, in particu-
lar the assembly of the pitch control swashplate which
is one of the most stressed and vital elements of a
helicopter.
Let us examine certain problems in the calcula-
tion of radial-thrust ball bearings, with considera-
tion of misalignment of their races under load. The
results obtained in solving these problems yield an-
swers to the basic questions arising in the designing
of bearing assemblies for helicopter units which have
to absorb large moments .
Fig .4*16 Diagraon of
Loading of Two Ball
Bearings by Radial
and Axial Forces and
Moment.
Let the bearing assembly, consisting of two
radial-thrust ball bearings, absorb a combined load in
the form of a radial force R applied in the middle be-
tween the supports, an axial force A, and a moment M
(Fig.4»l6)* It is assimied that the force R and the
moment M act in one and the same plane.
Let us assign the index 1 to that bearing of a given assembly for which the
pressures on the balls caused by the action of the force R and the moment M are
cumulative. A3J. quantities pertaining to this bearing will be written with this
index. The index 2 is given to the second bearing in this assembly and to all
quantities pertaining to it.
Let us direct the axes of the coordinates for the bearings 1 and 2 as shown
in Fig. 4*16. It is obvious that, in the coordinate system Xiy^zi, the force R
and the moment M always have positive values whereas the axial force A can be
either positive or negative.
The conditions of equilibrium of the shaft to which the bearings are mounted
reduce to the following system of equations:
405
iH /\-i — ■**2>
M=M,+M,+R,-^ + R,-^
(3.1)
where L is the distance between sipports.
Since the moments M^^ and Mg at small distances "between the si:ipports are not
only commensurable with the moments R^ — r- and Rg but may even appreciably
exceed them, eqs#(3»l) should be solved simultaneously with the equations of
static equilibrium of the bearings 1 and 2.
We assume that, in the loaded assembly, the center of the stack of inner
races is displaced in the direction of action of the forces A and R by distances
s and u and that the common axis of rotation of these races is turned in the /36Zi.
direction of action of the moment M through an
angle ^. The angle, z^ is the angle of misalign-
ment (obUquity) for the bearing 1 and for the
bearing 2«
The relative displacements'"" determining the
position of the centers of the inner races of the
bearings 1 and 2 in the coordinate systems Xiy^Zi
and XgygZs* according to Fig.4»17 are equal to
Fig .4 ♦ 17 Diagram of Dis-
placements of Inner Races
of Bearings under the Ef-
fect of an Arbitrary Ex-
ternal Load#
(3.2)
Here,
^pr
c =
2Cr
= half of the axial preloading with
which the bearings are installed;
= ratio of the interbearing distance
to diameter at which balls are lo-
cated; e = ^ 3:*o/g*
After writing the equalities (1.16) for both bearings and substituting
eqs.(3.2) into them, easy transformations will give
"''" All relative quantities, as before, are expressed in fractions of the distance
406
1
/to7pi -Vtan p2
1
+ C
{fan^<2 + C) Xifi {tan^i ^ C) ^2^2
cosPi
^f tan^x-Vtanh ^^
I COS Pi COS p2 J
COSP2
(3.3)
In bearing assemblies of the type in question, we generally use radial-
thrust ball bearings with large initial contact angles, for which the relative
displacements 61 and 63 rarely exceed 0#25 sin^ Po* ^or the indicated values of
61 and 62, the equalities (3-3) can be replaced by the following approximate re-
lations :
1
2 cos [
(X181 — X282);
1
2cosPo(^^^Po+ Q
■(Ml + V^2).
(3-M
For the selected direction of the axes of the coordinates, the angle |q de-
termining the position of the most loaded ball in the bearing 1 is always equal
to zero. The angle t|ro2 characterizing the position of the most loaded ball in
the bearing 2, depending ijpon the ratio of the radial force R to the moment M,
may have either a value of zero (for the prevailing moment) or may be equal to
1S0° (for the prevailing radial load).
Bearing in mind the latter circumstance, it becomes possible by means of 7^65
the equalities (1.2^), (1.25), (1.2l), (1.27), and (3.4), taking the comments
made on the order of the quantities 6 -l and 63 into account, to represent the
forces and moments taken by the bearings 1 and 2 as follows:
^-=«sinPoyu(l+^n-yf^ + c„— 5^
^02
2COS%2
z\d
i^=^,SlfcOsPoy2l(l-^2/^^^?0-7^~^2/^''^?^ '''
1 +Xi
■^21
I + X2 cos 4/02/ '
S02
l+Xj ^ ' I + X2C0SW'
i'Vi^ y 14-A2COS702 1 + *•!/
'6a
-By=Bolli' cos PoAs h-b22t^' % ■ ,— T^
ZVdt^ \ 1+ A2''f>« tLr-
X2 cos 4^02
-c.M^o-^)
COS li'oj;
M2
'•o'^'^te
2-=5o8o'2sinPoyj2(l+622--
^02
H-X2COS<1'02
1-22
801 \
i+V
COS<J'o2.
(3.5)
407
The foUowdiig notations are adopted in eqs.(3*5)
Jn
hi .
sin 2po (tan^o
wpo + C) J
Cu =-
hi sin2Po(fan?o + Q
J2I
sin2po(/iwpO
bi2 = (^o^% +
J22
J12
sin2P{r(/JwP(
^"^1
^2 cos %2\
.722
Cio='^^^- cost['o2;
^22 = ^^f'Po+'-^
732
7*22
1
sin2po(Awpo+0
■ 1 U2 COS tpo2;
^22 —
__Jz2
sin2Po(fa/7po + C)
COS
^'
(3.6)
As follows from eqs.(3«2) and (1.13), we have
2A,
tan^,+tan^2='^=2tan%-r—r'
cos po COS Po
Since, at 6" ^ 0.25 sin^ Po^ we can put approximately tan p = tan p +
6
the last expression -will yield
sin Po cos Po
Si + 82 = 2Ap,sinPo.
Using the equality (1.21), we finally have
^01
c>02
1 -f Xj 1 +X2COS tp
'02
= 2A sinSo.
7366
(3.7)
The relations (3*5), (3-6), and (3.7) together with eqs-(3'l) make it pos-
sible to determine all parameters characterizing the performance of radial-thrust
"ball bearings for closely spaced supports, when misalignment of the races under
load cannot be disregarded. As shown by numerical calculations^^ the accuracy of
these relations obtained on the assumption that the quantities 61 and 63 do not
exceed 0.25 sin^ Po a-"t initial contact angles of Po^ 26^, with which we usually
deal in bearing assemblies intended for absorbing large moments, is sufficient.
We will next analyze the basic calculation cases encountered when designing
bearing assemblies of this type for helicopter units.
408
2. Case of "Piire^ Moment
If a bearing assembly consisting of two identical radial-thrust ball bear-
ings absorbs a "pure'-' moment (Fig .4.18), then by virtue of the identical loads
on both bearings we must have R^^ = Rg, A^ = Ag, and M^ - Mg* It is logical that,
in this case, i(ro2 = ilfoi = 0* ^i = ^2* ^^ ^01 - ^02»
€'9u
300""
2^0"
180'
-
n
§
/
— ^- /
l^l
/
"^r
/
(/
/
I
/
/ ,
/ 1
j
/
/ y
!
^
^
Ia^
1
^
r
•
D a 01 a 02 0.03 /SprSinJ^g
Fig. 4 •IS Diagram of Loading of Two
Ball Bearings by a "Pure" Moment.
Fig .4.19 Effect of Preloading on
the Loading Zone.
As indicated in eq.(3«7)j at "^q^ = 0, \i = Xs, and 60;^ = ^02^ ^® have
^01
1 +Xi
^sinpoA
(3.a)
Consequently, under the effect of a "pure" moment.
sin Pq Apr
(3.9)
It is understandable that the ratio
'oa
should always be greater
sin BoApr
than unity. This becomes obvious when taking into account that the product
sin PoS*pr represents the relative convergence of the raceways caused by the pre-
loading, i.e., the relative approach of the raceways present before applying /367
an external load to the assembly.
Equation (3 •9) determines the loading zone as a function of the level of
the load and the preloading. This relation, in particiilar, shows that, to have
all balls share the load, the bearing should be mounted with a relative preload-
— ^01
409
The effect of the preloading on the loading zone is shown in Fig . 4*19 •
Frequently, the preload is not given as a relative axial displacement 2Apj.
but as a corresponding axial load Apj., deterniined "by the expression
.,=.^.wKfs,n..fe(l+|Si.i^. (3_^j
Since, under the effect of a "pure" moment, we have R^ = Rg ^^ ^i = -A-s,
the first two equations of the system (3*1) are identically satisfied. The
third equation of the system, for the case of a "p-ure" moment, can "be transformed
by means of eqs.(3»5)* (3*6), and (3»7) in the following manner:
X
r^-2^oSorsinPo(l+CcotPo)y2iX
(3.11)
1 + C cot po I 721 1 + C cot po
Using eqs.(2.3), (2.5), (3*9), and (3*11), it is easy to construct the
graphs of the relations ^^=Fq ( ^ A and -%- = /;/ — ^\, from which we can
find the maximum and equivalent pressures on the balls- The curves shown in
Fig .4.20 can serve as a typical exanple of such graphs. They were obtained on
the assunrption that Pq = 36^, C = 0, and Apr sin Pq = 0.01.
Let us now represent the maximum pressure on the ball P^-i^ in the form
°^ ~ 2J2ij^_ Joz sin po (T+ C cot%^ " 2roirsinMl + CcofPo) ' (3-12)
Accordingly, we put
p ^^p = 2,S7k(^^M
^?^ ^^ 2roZsinPo(l+CcoiPo) ' (3.13)
Where k^"^= ^^^ • k^"\
0.587 ^
It is necessary to note that under the effect of the "pure" moment^Pos = Pqi
and F^^2 = ^eq i-
It is easy to demonstrate that the coefficient is
(3.34)
Ho -
h\
"^o 1+C coX po 1^ hi l+Ccof Po 7 1 + ^1
410
At zero preloading when Xi = oo according to eq.(3*9), we obtain from /%S
eq.(3.14)
/c(A^) =
1
1 +
0.868 (
1 — Cfa7yPo >|2
H\
(3.15)
The values of the coefficient k^"^ corresponding to eq.(3.15) can be deter-
mined from the graphs in Fig.4«2l. Here, the abscissa gives the quantity M =
1 M _ ^.^ , 1-C tan Po I . ^ ,
The quantity p = | ^^^ ^ — ^— p- \ is taken as
zvd^a sin Po(l + C cot Pq) ^°
a parameter.
tan 3o + C
1.6
1,0
0,8
0.6
Poi
/
f
\
/
p,f
"
/
\'
r"
7
/
/
/
/
/
/
/
/
y
/
at ox 0,3 0,'t 0,5
M
Fig. 4. 20 Values of
vdL
-^^L-i — as a Function of
and
M
vd'
ba
for Po = 36°.
roZvdt.a
\0
0.9
0.8
OJ
(
i
. .
1 i
P=0.5
^
■-1
^^
^^£:io.
0^
t,2 M
Fig .4.21 Values of the Coef-
ficient ko"^ as a Funct^ion of
the Parameters p and M.
The graphs in Figs .4.22 and 4.23 show the mode of variation in the coef-
ficients k^"^ and k^"^ as a function of preloading when Po = 36° and C = 0. The
curves j.ndicate that the preloading should be selected such that the parameter ^x
lies within the limits of 1 to 1.25* At such a selection of the preloading, the
coefficient k^"^ and hence the maxmum pressure on the ball drop by 10 - 12^. In
this case, the coefficient k^ "^ and, together with it, the equivalent pressure on
the ball will keep approximately the same magnitude as in the case without pre-
loading. Analogous conclusions can be drawn from a study of other combinations
of the quantities Po and C*
It is extremely important to estijuate the effect of preloading on the angu-
411
lar stiffness of the bearing assembly • This is easily done "by means of the
second equation of the system (3*4) which, for the case of a "pure" moment, can
"be represented in the form
e=z
cospo(/2wPo4-C) 1 +Xi
(3.16)
Equation (3*16) shows that a change from Xi=ootoXi = l- 1.25 leads to a
decrease in misalignment of the races of the bearings by a factor of 2.2 to 2.
It is obvious from the aforesaid that, in installing radial-thrust ball
bearings with an optimal preloading corresponding to values of the parameter X^
from 1 to 1.25^ the service conditions of the bearing assemblies loaded by a /369
moment iirprove noticeably.
0.02 O.O^f 0,06 OM l\pr^inpQ
K-g.4.22 Values of the Coef-
ficient ko"^ versus Preloading
for Certain Constant Values
of
M
roZvdt,a
OM OM 6^sinpo
Pig .4. 23 Values of the Coef-
ficient k^"^ versus Preloading
for Certain Constant Values
of
M
rozvdt,
The preloading at which the parameter X^ = 1 to 1.25, in practical calcula-
tions, can be conputed from the approximate formula
A..=
(1.96 - 1.94) iVf
where
// = -
2-2.25
P' (2 --2.25) sin Po 1 1 + // [(1 .96 - 1.94) M]^/^
1 2/3
I '
(3.17)
l + Ccot Po
" 1 4- (0.905 — 1 .08) ^-'^f'^^o ]
. ' ' l + Cwf PoJ
In this case, the quantities Pqi and Pcq i are respectively equal to
p (3. 92- 3.88 ) /M _ »
" 2ro*slnPo(l+C<:orpo){H-//l(1.96-1.94)ATps) ' I
/»,^i =(0.657—0,645) />oi. J
(3.18)
412
3» S imult ajie£us_Actionj^^ and Axial Force
Z220
In the presence of simultaneous action of moment and axial force (Fig .4 -2^),
the loading conditions of the "bearings 1 and 2 are dissimilar, which greatly
con^licates the calculations for determining the "ball pressures. To find the
quantities Pqi, P^q i^ ^os* ^^ Pp.qs^ ^ number of auxiliary graphs must he con-
structed* The sequence of plotting such graphs is easily understood from the
following exairple:
let the initial contact angle te 0^^ = 36°. For siirplicity, let us assume
that the relative preloading is Apr= and the relative base C = 0, i.e., let us
study the case directly related to calculations of the bearings of the pitch con-
trol swashplate, for which these assunptions are sufficiently vaUd.
0,1 OA 0,6 0.8 10 ;.2 3f
Pig. 4. 2^ Diagram of Loading Two Fig .4-25 Typical Graph of Sq^ -
Ball Bearings by Moment and ibcLal = ^oiC^* ^) ^^r R=Ri-R2=0.
Force .
According to eqs.(3.5) for a radial load of R = Ri = Rg = 0, the quantities
^oij "^02^ ^i> ^^ ^3 ^3?e correlated t)y the following relations:
'■=• - ihP''^'"' K'"^''^' -(!tT ■''^^'^] rr^-
4-to2 po
/• c -/M
/ 22^^22
Sq'2
(3.19)
1 +h
In the equality (3.19), as in all subsequent relations, we have taken into
account that - during the simultaneous action of moment and axial force - we
have ^02 = ^oi = 0* J^"^ ^^ in the case of action of a "pure" moment.
From eq.(3.7) at zero preloading we obtain
413
OOl
ti02
1 + Xi ' 1 + X2
(3.20)
From this follows
p (1+^+1
(3.21)
Henceforth, the ratio
■'OS
will "be denoted everywhere by k-
Making_use of the equalities (3-6), (3*19), (3*20), and (3*2l), let us plot
the curves Sq-^ = ^oi(k, X^) satisfying the_condition R = Ri - Rg = (Fig.4»25).
Intersecting the obtained ciorves by lines 601 == const, we find the values of k /371
corresponding to the selected values of ^q^ for the given values of X^ (from co
to 0). Furthermore, taking X^ as a parameter, we can use eqs«(3»l)> (3*5)9 ^.nd
the preceding equalities for calculating the quantities
M
the ratios
•02
01
3/2 J ^eq 3 5/3 ^;
= K ^ and — = K^ —
P.
ZVd^a 3-oZVdta
-, and
■ eq 1
Wi
The results of the calculations are presented graphically, as is done in
A _ . M
Figs .4*26 - 4*28. Figure 4*26, from the given values of
and
rozvdf
permits determining the quantities 601 and X ^; when these are known it becomes
easy to calculate the maximum and equivalent pressures Pq 1 and Peqi* From
Figs#4*27 and 4*28, we find the ratios
■^02
Poi
and
• eq 3
• eq 1
and then calculate the
maximum and equivalent pressures Pqs and Peq2-
0.7
0.6
0,5
OA
0,3
0.1
OJ
The case Apj. = and Q = was
tlo^^y;%^o
•■-?
[h
y-
A
a
3
AB
—
—
—
7/
//
^
H
^
><j
"N.
M
■■/
m
^
^y
X,
v..
V
^
<
t/i
^
^
[><
^
\J
^
"vj
X
\
V
^Aro^z
<^
>
K
■\
N^
^
<
N
N
><^
K
x^
N.
N
^
y^
N
^
k-
'^ye->. ^
^
N
k-
0,1 OA 0.6 0.8 to
Fig .4 .26 Dependence of
U
M
2vrfJ,
rozvd^
on
ba
— , for Certain Constant Values
of 6q3_ and X ^ •
z^Jdta
analyzed above. For arbitrary values
of these quantities and also for
other initial contact angles Po^ de-
termination of the pressures Pqi,
^eqi* -^02 J and Peq2 ^s made in the
same manner as in the exanple under
study. It should be remembered that,
in the presence of preloading, the
quantity 6qi cannot be less than
sin PoApr •
_ We should note that the case of
App = and C = is characteristic
not only for bearings of the pitch
control but also for many large-
diameter roller bearings used in
rotary devices of modern machines and
mechanisms •
414
It is olDvious from the presented material that, for the prevailing moment
when Xi > 1, the hearing 2 is usually the most loaded, although at first glance
the service conditions of the hearing 1, toward which the aixial load A is di-
rected, seem more severe •
The presented method of calculating radial-thrust "ball hearings under the
combined action of moment and axial force requires a large volume of calculations
and constructions • Therefore, its use is warranted only in special studies hav-
ing the purpose of determining the peculiarities of the load distrihution in /372
hearing assemblies with closely spaced st^^ports, and also for plotting auxiliary
graphs for calculating individual standard structures. If such graphs have not
been constructed beforehand, the engineering calculations should use the sinpH-
fied procedure based on critical relations obtained for the case of small loads,
when the forces are distributed between the balls in the most unfavorable manner.
Pe^
Pe,!
-
..
"^
to
0.8
0.6
^^
^
—
._
-
QM
-
1 M
0.08^
'SorO.IO
m
w
—
y
-
Q QM 0.8 n 1.6 2.0 2M 4
Fig. 4^27 Dependence of the
Ratio
°^ on -^, for Cer-
tain Constant Values of 6oi
Fig.4-2S Dependence of the
P-n=. 1
Ratio
■•e q 2
' eq 1
on
-, for Cer-
tain Constant Values of Sqi'
4. limit Dependences on Small Loads
In the presence of small loads, it can be assumed that the contact angles
of all balls are approximately identical and equal to Pq .
After discarding in eqs.(3.5) all terms_that allow for the variation in con-
tact angles and substituting the quantities 6oi and 6^3 by the maximum pressures
on the balls Pqi and Pqs* which is more convenient for small loads, we have
^1 =^^^01 cos 30721;
^1=^^01 sin Poy'n;
Mi = ro2;PoiSinpoy2i;
/^^=iZpQ2 COS %J22\
M2^rQzPQ2Sin%J22.
(3^22)
415
Substituting the dependences i3»22) into eqs.(3»l), we obtain
R=zPqi cos ?o (y'si — ^^^V22 COS tos); ]
^ = /-o^^oiSinPo(l+^^°^ Po)(/21 + ^'^V22COS%2), J
(3.23)
6
02
f ,S21
2/3
where, as before, k = -= i —
^01 ^ Poi ''
Since in this case the angle |o2 need not be equal to zero, the equality
(3.7), under consideration of this circumstance, will yield
X2COS6q2 =
1 +
x(l + Xi)
1-(1+Xi)
2Ap,sin^
(3.2^)
Let us examine the system of equations
^2COS^'02=~[^(M-M+1 .
zm
(3.25)
It is easy to demonstrate that the values of X^, X^, and k for zero pre-
loading, satisfying equations (3*25), also satisfy eqs#(3*23) and (3.2f^), if we
set
/■o/?(l+Ccot Po)ta npo ^.
%z=rj
ro/?(l+Ccot Po)ianPo ^
(3.26)
M
, ro/?(l +Ccot Po)Un3o ^ .
^^^ M ^^'
M
(3.27)
ro/?(l + Ccofpo)^a;Tf
It must be recalled that, in the first case, the angle ^02 is equal to zero
and in the second, to 180*^ •
416
If the axial force plays the major role in the external load, then the
pressiires on the balls are usually* "written in the form
^01 — .~ — »
z sin^o
^'^^~ z sin So '
4^^A
02-
H.o^'
AA).
''?^" zsinpo
(3-2S)
If the moment predominates, it is generally acceptable to use the form of
notation given earlier:
^01 =
^,.=
4.37A^^f^>M
2/-o-2'sin?on + CcofPo) *
2/-o-?sinPo(l + Ccof po) '
2ro2sinPo(t-i-Ccof&o)
2. 574"^ J ^T
2ro-z'sInPo(l + Ccof po)
(3-29)
According to eqs.(3*23), we have
/31k
f^(M) = ._ t .
°' 4,37(;2i4-x^'=^y22costo2) '
f,m) _^3/2^(f);
^3/2
'/12 *
^02 ==^- ^01 •
(3-30)
As regards the coefficients k^/^ , k*^/^ , k^"^ , and k^'^^ , these are equal to
fC{^)'-
0.587
^(f); K^^W.Klf^; ^2^^^==^-^ '^of ; '^u^2^> = ^242^^
It should be noted that, between the coefficients k^^^ and k^"^ , there ex-
ists the following relation:
'^^^'^^^r
M
rQA{\+Ccot?o)
(3.31)
417
Fig .4*29 Nomograms for Approxijnate Calculations of Bearings
Loaded by Axial and Radial Forces and Moments.
The solution of the system (3*25) can loe represented in the form of graphs
shown in Fig .4*29 • From these graphs, knomng t and v, it is easy to find the
values of k and X^., from which we calculate the product^ X^ cos jfos f-^ then /375
calculate the coefficients
^01
^1 9
^0 2 9 ^2 9
or k
01
^1 9
^03
,(M)
After this, it is not particularly difficult to determine the pressures on the
balls •
The graphs in Fig*4»29 are interesting in that neither the angle P nor Q
figures in them. Thus, we have arrived at a rather convenient approximate method
of calculating radial-thrust ball bearings with large initial contact angles in
the most common case of their loading. The preloading, as already demonstrated
by a study of bearing assemblies loaded by a "pure" moment, mainly has an effect
on the stiffness of the system but leaves the calculated values of the maximum
and equivalent ball pressures practically unchanged. Thus, the presented approx-
imate method of determining these quantities, based on the assunption that the
preloading is equal to zero, can be used for solving a rather wide range of
problems associated with the calculation of radial-thrust ball bearings with
large initial contact angles installed in bearing assemblies^ with closely spaced
supports and absorbing an arbitrary combined load. In sinpler loading cases,
when examining the critical distribution of forces between balls corresponding
to small loads, it is relatively easy to allow for the preloading if necessary.
418
Let the assembly be loaded only "by the moment and the axial force • In the
absence of a radial load, as follows from eqs.(3^23)> we have j^i - ^^^
J22 cos ^02 - 0» Making use of this relation, we reduce the expressions for the
coefficients k^^^ and k^^^ to the form
"'ni - » "Til
"01
4.37/21 "^ 2/21
''«^'- 4.37/22 ' ''''^''2L '
^ 2.57/21 ' 2/21
2.57/22 2/22
(3.32)
Equations (3*32) are valid also in the presence of preloading.
The coefficients k^oi^ . 1^2^ , ^i"^ , and k^g"^ determined from eqs.(3.32) for
zero preloading can be found from the graphs in Figs#4»30 and 4.31* Since, at
small values of the ratio , the coefficients ko ^ and k^"^ for the bearing 2
T
are substantially greater than for the bearing 1, the question naturally arises
whether we can equate their values by proper selection of the preloading* With-
out dwelling on the transformations related with the solution of this problem,
since they are sufficiently obvious from the foregoing, we will directly give the
the final solution. Figure 4»33 presents curves giving the values of the ratio
— HL_ E2_ at which identity of the static and dynamic loads of the bearings 1
and 2, i.e., equality of the coefficients k^oi^ and k^^^ or kV^^ and k^"^ , can be
theoretically secured. The values of these adjusted coefficients are shown in
Figs .4 '30 and A- '31 as broken Unes.
For assemblies which should have high rigidity, it is desirable that all /376
balls in both bearings be loaded. This problem is also easily solved by proper
choice of the preloading. Since, under the combined action of moment and axial
force, the loading zone in the bearing 1 is always greater than in the bearing 2,
the condition of conplete loading of the balls of both bearings is the inequal-
ity ^3 s 1. The values of the coefficient 4"^ , k^02^ » ^i^^ > and k^g"^ and the
ratio — 3^^^-= —, corresponding to the case Xq ~ 1^ are also given in Figs .4*31
and 4*32. Figure 4*33 shows that, with a preloading which ensures loading of all
balls of both bearings, the angular stiffness of the assembly increases by a
factor of more than 2, which causes the maximum and equivalent pressures on the
balls to increase by about 10 - 15% •
The limit dependences obtained for the case of small loads are rather con-
venient for practical calculations, since they substantially lessen the labor-
iousness of determining the pressures on the balls. It must only be remembered
419
/M)
'0
7,2
10
<?
i^z-
1
..^<nO'
\
>^
<^
^
^^^^rr.^
^"
'^r^Q
X
0,8
0,6
^^.(M). , _/ 1
1
3
0.
*
0.
8
1.Z
±
r
Fig. 4*30 Values of the Coef-
ficient k^"^ = 4"^(— ) for
the Case of Simultaneous Action
of Moment and ibdal Force*
Fig .4.31 Values of the Coef-
ficient k^"^ - k^"V~^) for
\ T
J
the Case of Simultaneous Action
of Moment and Axial Force.
AprSm
h
^01
0.6
^r
1
OA
0.Z
s.
\
AMI
~
-
\
M
a
8
u 4
Fig .4*32 Values of Preloading
for Ensuring the Conditions
kl"^
= k:
(M)
^01
= k,
(M)
2
and
10
0,8
0.6
0,^
0.2
OA 0,8 tl 16 2,0 IJi ^
Fig .4 -33 Effect of Preloading
(Condition \^ =^ \) on Angular
Stiffness of the Assembly.
^
-.^
.^?r-n
■—
^^
^
"^
/
^__
~^~-
■
Xp = 1.
that the use of these limit dependences leads to a certain overestimation of the
rated "ball pressures. For contact stresses of the order of 20,000 kg/cm^, /377
this amounts to I5 - 25% for the angle Po = 26° and to 2 - YJ% for the angle
Pq = 60°. Allowing for this fact in calculating the rated loads of a given "bear-
ing assembly with the use of the above critical dependences, the values of the
safety factor can be reduced considerably.
On the basis of the dependences presented above, we can determine the pres-
sures on the balls of eccentrically loaded double-row thrust ball bearings. Here
we must remember that for an initial contact angle of Pq = 90° at Apr = 0,
eqs.(3.32) yield the "exact" values of the coefficients entering eqs.(3^28) and
(3*29) • We note that in this case the quantity t represents the relative eccen-
tricity
(Fig .4 .34).
420
If a single-row thrust l^all bearing takes an eccentrically applied axial
I. then
force, then
where
(3.33)
The values of the coefficients ko'^^ and k^'^^ are found from the curves
plotted in Fig .4*35 • These curves are obtained from the values of X correspond-
ing to the equation
^ = T which, in turn, is obtained directly from the con-
Ji
ditions of static equilibrium
Fig .4.34 Double- Row Thrust Ball
Bearing Loaded by Axial Force and
Moment (Eccentrically Applied
Axial Force) .
T !
;
5
'-
-~
'"rh
U
—
- —
~i/.
3
—
--
y^f/
2
1
"
: i
( i
1
0.2 Q^ 0,S 0.8 T
Fig .4 •35 Values of the Coef-
t k^^^ and k^'^^
FiHiction of T .
ficient k^^^ and k^'^^ as a
5. Distribution of Load between Rows of Balls of Double- Row
Radial-Thrust Ball Bearings
Radial-thriost ball bearings with initial contact angles of 26 and 36°,
having a small preloading, are widely used in helicopter conponents designed for
taking simultaneously acting radial and axial loads (Fig.4«36).
We will attenpt to establish the manner in which the load is distributed
over the rows of balls of such bearings, working under conditions precluding the
possibility of a noticeable misalignment of their races •
Keeping in mind that a small preloading has little effect on the ball /37B
pressures, we will use the limit dependences given in the preceding Subsection
421
Plg.4^36 DoulDle-Row Radial-Thrust
Ball Bearing Loaded "by Radial and
Axial Forces.
^*^^
^/
-1
r'
/
^^
th rows
/]
of balls y
operativjy^
/
One row
of balls
^
^
operative
1 1
f i'-KS? 1
±
T
Fig •4.37 Values of \^ and of the
1
Coefficients k^^^ as a Func-
tion of
for an approximate solution of the problem.
Setting e" = and Ap^ = in the equalities (3*4) and {3*l) and recalling
that 6t =
'ox
and 6p =
^02
-, we find Xg = X^. The index "1" is
1 + Xi 1+^3 cos llTos
given to the row of Iballs toward which the axial force is directed. Since, if
the bearing is loaded by radial and axial forces, the angle will be ^q^ = 180°
and thus cos ^o^ = -1» so that eqs.(3.23) will yield, for the case under study.
^ = 2:^01 cos Po
l\ U3/2 . 1
(3 '3k)
(3.35)
The dependences determining the pressiares Pqi and Pq^ for double-row radial-
thrust tall bearings can be written in the following manner:
01 ■
_ 4,37
4?>R
z cos
2
^01.
where
«(f)=
..37[...e-;^f,.]
(3.36)
422
to
The equivalent pressures for both rows of balls are, respectively, equal /379
(3.37)
The values of the parameter Xi and of the coefficient k^^^ as a function of
the quantity = -— cot Pq can be found from the graphs in Pig. 4*37-
T R
Pig.4»3S Pitch Control of HeUccpter Rotor*
As shown by calculations, the first row is always more loaded. For ^
T
s: 1.67 at A.1 ^ 1, this row carries the entire load applied to the bearing. We
note that, for t =0.6, the ratio — — - = I.67 tan Po •
R
In the presence of predominantly axial loads, when one row of balls is
operative, more acciorate results are obtained by using the dependences in Sec-
tion 2 for calculating double- row radial-thrust ball bearings of all types, inr-
eluding those examined in this Subsection.
423
6. Examples of Calculation
Example 1 « Let us determine the rated life of the "bearings of the swash-
plate of the pitch control (Fig.4«3S) loaded loy the moments M = I50 kg-m and
rotating at ^0 rpm* The bearings have the following parameters: Po ~ 36*^,
dba = 9.525 mm, z = /f2, ro = 79 inm.
Relative base C = 0.1, preloading Apr = 0. /360
Since the bearing has zero preloading, the coefficients ko ^ and k^"^
needed for calculating the loads on the balls are determined by means of Fig.4*2l»
We then calculate the quantities
Q =
1— CAwPo
A?/?Po4-C
M =
1 — 0. UP. 726
0.726 + 0,1
1 M
= 1.12;
-?vrf^^sinpo(l+Ccof Po) ro
' ''' .0J44.
42>1).9.5252m0. 588(1+0.1x1. 376) 0.079
Since — ^^ Cout) ^ 0.515, the coefficient v is taken as equal to unity.
According to Fig .4*21, the value k^"^ = 0.912 corresponds to the obtained
values of p and M.
Thus, the maximim pressures on the balls in both bearings will be
p =.p ~ 4.37/c^^)^ _
^^ °^ 2ro^sinpo(l+Cf(?rpo)
^ 4.37x0.912^50 =134 7 k
2*0.079^42^0.588(1 + 0.1x1.376)^ * ^'
In the examined case, X^ = Xg =00, Consequently,
/2^j=iD^2 = ^x=«Poi=0-587»134J = 79.1 kg.
The equivalent pressure P^q g, , determining the Hfe expectancy of the as-
sembly, can be found from eq.(2.1j45. Taking into consideration that, at X^ =
= Xs = CO, kjci = kk2 = ^•2 (see Fig.Zf.*5), this equation -will yield
1
/".p =2'-'"'fKi^, 1 =51.21 « 1 .2-79. 1 = H4.8 *p.
10
Here, we assume that * = q
42i».
For bearings with the indicated dimensions, according to eq.(2.42) the co-
efficient of utilization will be
C^-eSx^o-^ ^ ^^- 65x420-7 X ^i^?5? =67 794.
l + 0.02fifbA 1 + 0.02x9.525
As a result of stand tests, for "bearings of the pitch control cam plate the
product of the coefficients k^jk^kw = l.l.
In conformity with this, eq«(2.43), considering that h - hio, will furnish
(nhf-^=9 = ^^^^ ^40.5.
^ 0.39xfH,^CtK^^zPet^^cos<?Q 0.39^1.1^42x114.8x0.809
Hence,
/zA = 228 370*hd /^ = ^?M^ j:tj 950 hfi
240
Ebcample 2* Let us calculate the maximum and equivalent pressures on balls
in bearings examined in exanple 1 when they absorb a moment of M = 60 kg*m and
an axial force of A = 500 kgf .
Since the relative base Q is small. Figs. 4*26 - A-»2B will be used for /381
determining the indicated pressures .
From the quantities
M 60
rozvd]^ 0.079x42x1x9.5252
and from
= 0.1993 ^i^"^2
^^^ 0.1312 k(j/mm2
zvdl^ 42). U9. 5252
by means of Fig.4»26 we find Sqi = 0.055 and Xi = 0.7«
Furthermore, let us calculate Pqi and Peqi* Since Xi = 0.7 corresponds to
the value w = 0.67B, then according to ecp.(2.2) and (2.6) we have
Poj = ^oVfi^2B^'^ = 62)clx9,5252K0.0553/2 = 72.5 k^ ;
^9i = 'Z2'^oi=49.2k3r.
From the graphs in Figs .4 '27 and 4»2B we find the values of Peqg/Peq i ^^^
P02/P01 • Using these values, we obtain
/>02 — 1 -085/^01 =78 J kg;
^^2^0.875^^1-43 k^.
425
Section 4- Calculation of Tapered Roller Bearings under Combined loads
"bearings at^sor'bing combined loads*
Cross section at angle ^
to plane of loading xOi
1. Calculation of Single-Row Tapered Roller Bearings
Methods were presented above for .calculating radial and radial-thrust "ball
let us now examine the peculiarities of cal-
culating* tapered roller bearings working un-
der conditions of a complex load*
First, let us give the solution of the
problem of determining the forces acting on
the rollers of a single-row tapered roller
bearing at given values of the radial and
axial loads applied to it (Fig.4«39)«
The normal forces P^ and Vl exerted on
the roller by the outer and inner races are
correlated b^r the relation
p._ cos(Y-Yt) p
COS Y^
(4.1)
Fig ,4 •39 Diagram of Loading
of Tapered Roller Bearing by-
Radial and Axial Forces.
For the usual values of the angles y
and 7^, we can consider for all practical
pixrposes that
p^= p^^
(4.2)
In conformity with the Hertz theory, it is possible to set, for the case of
Unear contact and with sufficient accuracy,
P^^Bl^, (4.3)
where 6^ is the convergence of the races in the cross section located at an
angle ^ to the loading plane.
In the absence of misalignment of the races under load, the convergence b^
is determined by the expression
7382
64, =5Sin p+ticospcosil).
(4-4)
Here,
u and s = radial and axLal displacements of the inner race relative to
the outer race, reckoned from the position at which the clear-
ances in the bearing are selected;
P = angle of taper of the outer race.
Having put
— cot M^.
5
(4.5)
426
we obtain from eqs«(4*3) ^-^id (4*4)
P4,=5 s sin p (1 -h^ cos i|)) .
(4.6)
If the direction of the radial load coincides with the positive direction
of the z-axis (R > O), then the displacement will be u > 0. In this case, the
center of the loading zone lies in the cross section ^Jr = i|ro = 0. If the radial
load acts in the opposite direction (R < O), then the displacement will be u < 0,
in which case the center of the loading zone is situated in the cross section
^ = ilfo = 180°.
According to eq.(4.6), the maxLmiam value of the force P^ is equal to
Po=5 5sinp(H-A,cosil)o). (4-7)
Using the equality (4.7), we finally have
-^ (1+Xcosl^).
1 + Xcostt'o
(4-8)
As follows from the conditions of static equilibrium,
Po
/? = ^ - cos p V. (1 +'^ COS ^) cos ^1>;
i= ^
-sinpy^(l + Xcosil)).
(4.9)
1 -t- X COS 4^0
For the usual niomber of rollers, eqs.(4.9) can be replaced by the relations
Here,
R=>PqZ cos pj2C0s%\
^10
•^'^^ o^n-u!.. I , f (l+>-COS^oCOS^)^^ =
+/.
2jx(1 +Xcos4/o)
[(+;-*/;) + 2^ <^os to Sin t;;];
%
yo = ~ ' r (1 +X cos to cos t) cos ^d^=
^2 2jt(l + Xcosto)J TO t; T T
^Ic
1
2rt(l+Xcos«l/o)
{2sin<l-;,+ l^ f(1';'-<^,;^+sin2+,;|},
(4.10)
(4.11)
427
The iDOundaries of the loading zone if ^' and i|r' are determined in the same /3B3
lo io
manner as for the radial and radial-thrust ball bearings [see eq*(1.20) and the
e:xplanation to it]*
Equation (4»10) will yield, if for sinplicity we set R > and thus to = 0,
Pn =
1 R
J2 JZCOS^
(4-12)
The equivalent pressiure' P^q for a tapered roller bearing can be repre-
sented in the form
a = ^^o.
(4.13)
where
w=-
1
1 + X COS 4^0
-l0.3
— { (1 -}- X COS % cos 4*)3'33^]>
The values of the quantities ji, jg, and w as a function of \ cos ^q are
given in Table k*l-
TABLE 4*7
X COS t}/o
71
J2
w
X cos tj^o
h
0.405
h
0.268
w
1
0.5
1
2.000
0.698
0.1
0.909
0.454
0.913
2.500
0.389
0.267
0.692
0.2
0.833
0.417
0.853
3.333
0.371
0.264
0.686
0.3
0.769
0.385
0.806
5.000
0.354
0.261
0.679
0.4
0.714
0.357
0.773
10.000
0.336
0.256
0,670
0.5
0.667
0.333
0.751
±00
0.318
0.250
0.660
0.6
0.625
0.312
0.738
-10.000
0.300
0.242
0.648
0.7
0.688
0.294
0.729
-5.000
0.281
0.234
0.634
0.8
0.555
0.278
0.725
-3.333
0.261
0.222
0.617
0.9
0.526
0.263
0.722
-2.500
0.240
0.210
0.598
1.0
0.500
0.250
0.720
-2.000
0.218
0.196
0.575
IJll
0.479
0.258
0.718
-1.667
0.194
0.178
0.548
1.250
0.460
0.264
0.714
-1.428
0.167
0.156
0.518
1.428
0.440
0.266
0,708 ~
-1.250
0.136
0.130
0.484
1.667
0.424
0.268
0.704
-1.000
0.000
0.000
0.000
Calculation for tapered roller bearings, just as for radial-thrust ball
bearings, is usually carried out by means of reduced static and dynamic loads.
These loads are found from the condition that
428
1 11 1 ■■■■■■II II
72
'X-
^^ ^2x-«^*^^^^
Qo ^ ^ Qo
zcosp zcosp
^-=2.64 '^
Z COS p
(4.14)
A conparison of the equalities (4-14) with eqs.(4-l2) and (4-13) yields /384
^; ) (4.15)
where .
72
x_oo 0.25
° h h
(4.16)
In conformity with eqs.(4.10), the^ values of the parameter X needed for de-
termining the coefficients k^'^^ and k^*"^
ne VaXUtJa uj. one pcuajiicoci A. ncc
^1^^ should satisfy the condition
(4.17)
Since we have — — = 0.5 at X = 1, the value of t should not exceed 0.5 for
J2
all rollers in a single-row tapered roller bearing to be loaded.
At T s 0.5, the values of the coefficients k,^"^ and k^"^ can be determined
from the graphs in Fig. 4. 40. These graphs were plotted on the basis of the
equalities (4.16) and (4.17).
If T ^ 0.5 and thus X ^ 1, the expressions for the integrals ji and jg take
the form
Jy
1
72=-
1 + X'
1 X
2 1 + X
(4.18)
From eqs.(4.17) and (4.18), we find
l-=2 + -L.
Thus, for T ^ 0.5 when all rollers share the load, we have
(4.19)
429
iff. 0.25 n c I 0-25
(4.20)
As follows from the equality (4»20), for all loaded rollers.
Q=0J6wR-\-0MwA<:ot p.
(4.21)
(4.22)
The values of w as a function of t are conveniently determined from the
curves shown in Fig .4. 41.
These relations fiornish an answer to all "basic problems arising in calcu-.
lating tapered roller bearings that take combined loads, provided the misalign-
ment of their races can be neglected •
f^f^' vt/^j
10
"'o"
1,5
)
y.
^
/
1.0
-
—
^
«<»'
_
0,5
0,6 OJ 0,8
0,8
0,6
0,9
O.l
OA
0,6
mi
Fig. 4. 40 Values of the Coef-
ficients k^o*^^ and k^^^ as a
Function of t .
Fig .4.41 Values of w as a
Function of t •
As shown by Fig .4.40, in the region t = 0.6 - 0.8 the curves of ko'^^
4^^^ (t) and k^*^^ = k^'^^ (t) have a rather well-defined minimum. This ii
.(R) -
^0
^o"'' Ct) and k"^"^ = k''"^ CtJ have a rather well-defined minimum. This indicates
that proper choice of the contact angle p, for a given combination of radial
and axial loads, will ensiire maxim-um and equivalent pressures on the rollers
having a minimum value. The optimal contact angles at which the conditions Pq =
= Po^"" and Peq = Y^l^ are satisfied are determined from the graphs in Fig .4.42.
These graphs were plotted on the basis of investigations of the relations
2Pn
= F,
(' ^
^ ^
R
to 30°.
and
zP.
R
R
= F f _^)^ for a number of contact angles in the range from
R )
430
2. Remarks on Calcu l ation of Bearitig Assemblies of Two
Tapered Roller Bearings
If a bearing assembly consisting of two tapered roller bearings is loaded
by a moment acting in combination with radial and axial forces (Fig.4»43), the
following system of equations can be used for its calculation:
;?=zPoi cos ? (721 —V22 cos %2);
A=zPoiSinP(yn — V12);
M =- rozPoi sin p ( 1 + C cot p) (y'si + ^722 cos ^'02);
^2^03^02 =
1 +
x(l + Xi)
1-(1+Xi)
2AprSin p
(4.23)
which is analogous to the system of equations (3 -23) and (3 ♦24) describing the *
conditions of static equilibrium of bearing assemblies with two radial-thrust
ball bearings, on the assi:imption that the contact angles of all balls are identi-
cal and equal to the initial angle.
It should be noted- that eqs.(4.23) are "exact" since, in tapered roller
bearings, the contact angles are actually constant and do not change under load;
these equations are vaUd in both absence or presence of misalignment of the
races .
In eqs#(4.23) the quantity k is used in place of the quantity k^^^ . Thisis
OS
explained by the fact that, for tapered roller bearings, we have — =
= = K •
'01
In the case of zero preloading, the values of X ^ and k satisfying /3B6
eqs.(4*23) are found from the graphs in Fig.4.44» The quantities t and v here
have the same meaning as for the radial-thrust ball bearings [see eqs.(3-26) and
(3.27)].
hp\
30
-
—
^
c
V
n min
•*
h
r
20
>.
^
— ^^
<^
K
10
i^
^
p^p,™'''
^
f
U
0.2 0> 0,6 OS
R
Pig .4 •42 Values of the Optijnal
Contact Angle as a Function of
the Ratio A/^*
Fig .4 •43 Diagram of Loading of Two
Tapered Roller Bearings by Radial
and Axial Forces and Moments.
431
From the found values of ^3^ and k, we then calculate the quantity X2
cos \lro2*
The maximum and equivalent pressijres on the rollers in the bearings 1 and
2 are determined from the e^qDressions
^oa =
2K<f>iM
4t>A
ro^sinP(l -f-Ccot p) z sin p
^42^M
02"
ro-^sinpcl +C cot p)
'^^^~~ /-0-?sinp(l + C ctftpy
^(J.2-
1.324^^M
z sin p
4^>>1
/■o^sinp(l+C cotp) zsinp
(4.2^)
Here,
K f ^) = . —^
°^ 2(y2i 4-^722 cos 4^02)
; <>=x<0
1
Jn—'*' J\2
01
f^iM) ^ _?5:i_ ^(Af) . f^iM) ^J^ f^(M)
0.66
"01 ' '"2
A:[^) = 'ZK^i^^f > ; A:^^> = "^2^02^
0.66 "«2
02
(4.25)
Setting 3 = 90° in the system (4 •23)^ we arrive at the following equations
describing the conditions of static equilibrium of double-row thrust roller
bearings :
^=^^oi(/n— V12);
Af = ro^Poi(/2i+V
); I
^'22). J
(4.26)
The methods of solving eqs.(/f26) are obvious from the preceding; conse-
quently, we need not further discuss these here.
If we assume h = in eqs.(4»26) they will take the form
Z288
(4-27)
Equations (4*27) characterize the load distribution in single-row thrust
roller bearings .
The values of the maximum Pq = 'k.\^^ and the equivalent P^^ = k^'^^ -—
432
h^.7 --'
^f^:i^^^.^ ^
/ML
Fig .4.44 Nomograms for Calculating Bearings Loaded
tj Radial and Axial Forces and Moments .
pressures on the roller satisfying eqs.(4*27) are conveniently found by means of
the curves k^o'^^ = ^h'^ (t) and k^^^ - k^^^ (t) given in Fig.4.45.
At X ^ 1, when all rollers are loaded in the bearing, the integrals ji and
js are determined from eqs.(4*l^)»
It is easy to demonstrate that in this case which takes place when t =
M
roA
^ 0.5, we have
X = 2x;
(4-2S)
Section 5. Calcul a tion, of Vibrating Bearings
In designing helicopters, the proper selection of bearings for the hubs of
433
the main and tail rotors presents appreciatile dif f iculties • These bearings, as
is known, operate xinder specific conditions of vibration. They do not fail "be-
cause of contact fatigue but as a consequence of local wear of the race tracks,
which has come to be known as "false brinelling". It is understandable that the
usual calculation methods for such bearings are inap-
plicable «
K^.^ffl)
f
J
3-
r-^
-^
>^
>)
Q.l OA 0,6 0.8 T
Values of
Fig .4-45
the .__
k^'^^ as a Func-
tion of T .
Coefficients k^^^
The properties of the lubricant have a substantial
effect on the performance of vibrating bearings. Prac-
tice has shown that reliable operation of many Soviet
bearing assemblies for helicopters is possible only
when using special oils and lubricants. Therefore, in
helicopter engineering special attention must be paid
to problems of selecting the lubricants for antifric-
tion bearings. This primarily pertains to bearings for
the axial (feathering; hinges of the hubs of the main
and tail rotors, which absorb appreciable axial loads
generated by the centrifugal forces of the blades.
and
The complexity of calculating the bearings of hubs
of the main and tail rotors Hes in the fact that the
relatively low rigidity of their basic components,
especially on heavy helicopters, may lead to noticeable
defor*mation of the races, which is difficult to take into account when determine
ing the forces acting on the rolling bodies. So far it has been impossible to
develop general calculation methods that would allow for the effect of all fac-
tors determining the load capacity of bearings in the hubs of the main and tail
rotors • However, available experimental data permit certain recommendations as
to the selection of permissible loads and determination of the life expectancy
of the most common types of bearings used in these complex and vital units. This
is the same for calculations of the bearings in the hinges of the pitch control
and control mechanisms of helicopters which, just as the hub bearings, operate /,389
under vibrations. Here, it is merely necessary to take into account that the
loads absorbed by most of these bearings have a dynamic character.
1. Characteristics of the Mechanism of Wear of Antifriction Bearings
under Vibration Conditions
Let us examine the characteristics of the mechanism of wear of antifriction
bearings in the presence of vibrations.
At small vibration anqpHtudes, when contact of the rolling body with the
races takes place only at some spots on the tracks, dents from the balls or
grooves from the rollers will form in the bearing which, as their svocfaces chip,
change to deep pitting (Fig .4*46). Failure of the rolling bodies in most cases
begins only after appreciable damage to the races.
An analysis of test results shows that, in the presence of vibration, the
wear of bearings is largely determined by oxidation processes and special lubri-
cation conditions in the zones of contact of the rolling bodies with the races.
434
Flg#4»46 Races of Thrust Ball and Roller Bearings
after Extended Service in the Presence of Vibra-
tions of Small i\inplitude»
In the contact zones there is intense fretting corrosion. The oxidation
products of iron formed in this case mix with the lubricant and produce a luiiqae
polishing conpound which causes rapid wear of the tracks. The rolUng motion
of the rolling bodies creates lubrication "barriers" ahead of the contact area,
while jets of lubricant aft of this area tend to fill the space behind the mov-
ing body ( Fig .4 -47) • If the lubricant is too stiff and does not have time to
fill this space immediately, the portion of the track directly adjacent to the
contact area is coated only by a thin film of lubricant. Naturally, at the in-
stant of change in direction, the rolUng body will pass this poorly lubricated
portion sooner than the lubricant will be able to reach it* This causes the ap-
pearance of pressure peaks leading to acceleration of wear at the periphery of
the contact area between rolling elements and races, where the change in direc-
tion takes place. At very low vibration airplitudes, when the contact areas in
the extreme position of the rolUng body overlap, disturbance of the lubricant
layer may be constant. In this case, the pressure peaks increase even more and
the Hfe expectancy of the bearings decreases noticeably. An increase in mo- /390
bility of the lubricant will inprove the service conditions of vibrating bear-
ings . Nevertheless, even when using highly fluid oils, the service conditions
of such bearings substantially differ from those of bearings rotating in a single
direction.
It is obvious from the aforesaid that, in bearing assemblies operating in
the presence of vi brat ion > oils rather than grease should be used in all cases
where this is possible relative to design considerations. When grease is used,
the load capacity of vibrating bearings drops steeply*
2* Lubrication of B l^hly_ Loaded Vibrating: Bearings in the Presence
of Small Vibration Amplitudes
Since the properties of the lubricant have a considerable effect on the life
expectancy of vibrating bearings, one can discuss the permissible loads for such
bearings only in conjunction with the lubricants used.
435
Helicopter bearings sulDJect to vibrations can be divided into two basic
groups:
1) Bearings in the hubs of the main and tail rotors, pitch controls, and
certain control elements operating at vibration airplitudes tp to 10°.
For these bearings, the total number of vibrations between two major
overhauls, during which they are replaced, usually amounts to less
than 10 million.
2) Bearings of the control mechanisms which execute a limited number of
vibrations (up to 100,000) with airplitudes of more than 20°. We
stipulate that no overlap of adjacent contact areas exists in this
case.
Practice has shown that bearings of the second groip will operate satis-
factorily on high-quality greases. This is due to the fact that appreciable
grooving by the rolling bodies can be permitted
on the tracks of such bearings, since their
performance is -usually limited to the magnitude
of the permissible moment of friction.
Lubrication
barrier
Contact
Lubrican t
jets
^•^ Region of podrj"^
lubrication
Fig.Zf47 Diagram of Lubrica-
tion in the Presence of
Vibration.
In bearings of the pitch control and con-
trol elements belonging to the first group,
the use of grease results in a noticeable drop
in load-carrying capacity; however, because of
design considerations this is an unavoidable
evil and the insufficient lubricating quality
of the grease must be conpensated by some re-
duction of the permissible loads. Since the
permissible wear of the tracks of the hub bear-
ings of the main and tail rotors is not great,
prolonged operation at high contact stresses is
possible only if oils with a certain complex of physicochemical properties are
used.
The life expectancy of vibrating bearings depends largely on the quality
of the seal of the bearing units. In the presence of faulty seals that permit
penetration of atmospheric oxygen into the assemblies and also at small lubri-
cant volume and large air volimie, the life expectancy of vibrating bearings de-
creases noticeably. A rather effective means for increasing the service life of
bearings subject to vibrations is pressiire feed of the lubricant and, especial-
ly, use of oil circulation which continuously supplies fresh unoxidized oil /391
to the contact zones and carries off products of wear.
Let us discuss in greater detail the problems of selecting oils for the hubs
of the main and tail rotors, since these problems are vital for helicopter engi-
neering .
Oils for the feathering hinges of the main and tail rotor hubs . As shown
in nimierous experiments, the bearings of the feathering hinges, which absorb
rather large axial loads due to the centrifiigal forces of the blades, are espe-
cially sensitive to the physicochemical properties of the lubricant. The oils
for such assemblies, whose service life usually determines the overall lifetime
of the main and tail rotor hubs, should meet the following basic requirements:
436
,?;^^_k..
First, the oils should not cause a step-ip in the oxidative processes
taking place in the contact zones •
Second, the oils should retain high flioidity in the entire operating
tenperature range and should provide sufficient oil-filn strength over
the whole of the contact area.
The perndssilDle viscosity level of the oil is limited also by the permis-
si"ble inagnitude of the friction moment of the feathering hinge. Based on our
experience with operating the Mi-1 and Mi-4 helicopters, we can stipulate that
- at minimum operating tenperature - the kinematic viscosity should not exceed
90,000 cenbistokes. Tests show that, in this
case, there is no noticeable increase in the
moment of friction and no decrease in service
Hfe of the bearings due to decreased fluidity
of the oil. It should be noted that MS- 14 oil
which works satisfactorily in the feathering
hinges of the main rotor hubs of the Mi-1 and
Mi-4 helicopters at tenperatures as low as
-25°C, reaches the indicated kinematic vis-
cosity at a teirperature of -20^0.
0.002
0.001
Thrust ball bearings
S3U
n=1^0 eye/ win
UOQO
8000
moo flkg
Fig.4»4S Friction Coefficient
of Thrust Ball Bearings as a
Function of Brinelling Mark
Depth in Races*
By virtue of the specific operating con-
ditions of vibrating bearings, selection of
the oils and greases for these units should be
based exclusively on test results during vi-
bration* The standard procedure of testing
oils and greases on a four-ball tester is
conpletely unsuitable here. The lubricating
quality of oils and greases for feathering
hinges of main and tail rotor hubs is prefer-
ably checked in thrust ball bearings, since they operate at higher contact
stresses* Experiments show that lubricating materials with optimum performance
in such bearings are also best for vibrating bearings of other types, including
thrust bearings with "slewed" rollers which are presently used with success in
the main rotor hubs of all series-produced Soviet helicopters, and also for
multi-row radial-thrust ball bearings used in the main and tail rotor hubs of a
number of helicopters in other countries. Since the load in thrust ball bear-
ings is distributed uniformly over the balls, each contacting region of the track
diu?ing vibration can be regarded as an independent test object.
An essential factor in testing oils and greases for bearings of feathering
hinges of main and tail rotor hubs is a proper evaluation of the condition of
the tracks. Even at very moderate contact stresses, brinelling marks made by
the balls appear on the track after brief operating periods. If the appearance
of such dents, regardless of their depth, is considered as a sign of incipient
failirre of the bearing, then bearings which otherwise might still operate reli-
ably for a long time miost be rejected. The ciirves in Fig.4.4S indicate the man-
ner in which the depth of the dent affects the friction coefficient of a thrust
ball bearing. At a brinelling depth of 7 - 10 |jl, the friction coefficient in- /392
creases by 30 - 40^. An increase in friction coefficient within such limits is
usually not perceptible in service. Therefore, the condition of thrust ball
bearings at a depth of the dent up to ID p. should be rated as "satisfactory".
437
Such a "brinelling depth can be permitted also in radial-thrust ball bearings.
Tests have established that the MS-20 oil is one of the best for vibrating
bearings* In conformity with this, this oil can be adopted as a standard for
estimating the lubricating properties of oils and greases intended for service
in the feathering hinges of main and tail rotor hubs. The results of testing
thrust ball bearings running on MS-20 oil are given in Fig .4 '49 as a curve of
the Hfe expectancy a = a(nh) establishing the relation between the contact
stress a and the product nh of the number of vibrations per minute and duration
of operation in hours* The tests were carried out at a vibration aiiplitude of
the revolving race of cpo ~ 4*5°^ frequency of n = 240 cycle/min, and oil-bath
teiTperature of 20 - 40 C.
tSkgjcm"
36000
3^000
3Z000
30000
28000
26000
zmo
5f?0* lO^W** 15^W^ ZO^W*" nk-^'hr
oj
0.6
0,^
0.2
"X
\
N
\
U-^^'f
\
K
11 1A K6
LS 1,0 1.1 ^ttst
Pig .4 '49 Curve of life E^spectancy
a = a(nh) for Thrust Ball
Bearings .
Fig .4*50 Curve of Distribution
^max
of the Ratio ^
He St
^tes t
For the values of the contact stresses determined by the curve of Hfe ex-
pectancy plotted in Fig .4*49* 98^ of the brinelling marks on the tracks have a
depth not exceeding 10 ijl .
It should be noted that a definite statistical relation exists between the
maximum At^^ and average At^ depth of the dents. This relation is established
^max ^max
by the experimental curve of the distribution F f ) of the ratio ,
which, as is seen in Fig.4*50, is close to the Maxwellian distribution often en-
countered in engineering.
An analysis of the Hfe e^cpectancy ciorve in Fig. 4 -49 permits proposing /393
the following regime of accelerated selection tests of oils and greases for the
feathering hinges of the main rotor hubs: duration 100 hrs, number of vibrations
240/min, vibration aiiplitude k*5^ f contact stresses 34,000 kg/cm^. This regime
permits coirparing the lubricating properties of the tested oil with those of the
MS-20 oil.
One must remember that, with an increase in testing time, the role played
by oxidative processes in the contact zones increases in importance. Neverthe-
438
less, preselection tests of oils and greases for feathering hinge "bearings can
"be carried out by the above accelerated program, since accelerated tests fre-
quently permit the immediate rejection of many sanples.
Under conditions of vibrations, the MS-20 oil possesses excellent lubricity.
However, it can be used only in si:immer« Dioring the winter, the MS-20 oil is
usually replaced by MS-lfj- oil whose lubricating properties are also conpletely
satisfactory. Since the MS- 14 oil solidifies at a tenperature of -30° C, it can-
not be used at lower tenperatures, which greatly interferes with wintertime
operation of helicopters • Replacement of the MS- 14 oil by general-purpose oils
with low pour points does not yield favorable results. Tests have shown that
the feathering hinges of main rotor hubs fail rapidly when operating on ordinary
low-congealing oils, just as when operating on greases. This problem must be
discussed in some detail, since the regiJLarity of this result has long been dis-
puted by certain specialists in the field of lubricants, which has handicapped
solution of the problem of lubricating the feathering hinges of main rotor hubs
at low tenperat Tires .
Experiments have established that oils used for the feathering hinges of
main and tail rotor hubs, at a teirperature of 100° C, should have a kinematic
viscosity of not less than 9-10 est. Increased wear of tracks as well as chip-
ping and destruction of the rolUng bodies are observed when working with low-
viscosity oils.
Low pour point oils of high viscosity in the positive teirperature range
generally consist of a low-vLscosity mineral or synthetic base and a high-poly-
mer thickening agent. In most cases, the thickeners and the base itself have
low lubricating properties. Therefore, special antiwear additives containing
sulfur, chlorine, phosphorus, or certain combinations of these chemically active
elements are added to such oils. In zones of high contact teirperatures, the
additives react with the siirface of the metal, forming films of sulfides, chlor-
ides, and phosphides of iron which prevent a direct contact of the rubbing
bodies and thus reduce wear.
According to data obtained with the standard four-ball test device, the lu-
bricating properties of low pour point thickened oils with antiwear additives by
far exceed the lubrication properties of the MS-20 and MS- 14 oils. Neverthe-
less, they are conpletely unsuitable for working under vibration conditions.
This is due to the fact that, londer the effect of antiwear additives, oxidative
processes are stepped up in the contact zones; these play a decisive role in
the mechanism of wear of vibrating bearings. Here one must also consider that
most high-polymer coiipounds used in oils of low poior point readily deconpose un-
der mechanical action, with the formation of polymers of lower molecular weight.
Deconposition of the thickening agent leads to a decrease in viscosity of the
oil. In bearing assemblies working in the presence of vibrations, the average
decoirposition of the oil usually is negligible. However, since only small /394
vol-umes of oil directly adjacent to the rolling bodies are subject to the mechan-
ical action, local deconposition with a consequent drop in viscosity in the con-
tact zones may reach appreciable magnitudes and lead to noticeable loss of
strength of the oil filia.
Contradictory results are often obtained when testing oils with antiwear
439
additives. This shows that testing of such oils should "be carried out on a suf-
ficiently large number of sanples and that their lubricating properties cannot
iDe judged by solitary favorable results.
It follows from the aforesaid that the oils for feathering hinges of main
and tail rotor hulDS should contain no antiwear additives or degrading thick-
eners. This explains the unsatisfactory service of such assemblies with ai^ of
the conventional low pour point oils, in whose development the alDOve facts were
not taken into account.
Guided by the above data on the performance of lubricating materials under
conditions of vibration, the AU-Union Research Institute for Petroleum and Gas
Conversion and Production of Synthetic liquid Fuel (VNII NP) has proposed the
low pour point oil WII NP-25 for the feathering hinge of main and tail rotor
hulDS (Ref.28).
The oil WII NP-25 contains a low-viscosity petroleum fraction with a pour
point of -67°C and a high-viscosity thickener distinguished by extremely high
mechanical and thermal stability. Under the effect of high tenperatures of
friction, the petroleum fraction in the contact zones may evaporate; however,
contact of the rolling bodies with the races cannot take place because of the
presence of a film of the thickener which has relatively high adhesive proper-
ties. The high thermal and mechanical stability of the thickener and the oxida-
tion inhibitor only negligibly changes the properties of the oil WII MP-25 dur-
ing service.
The basic properties of the oil WII NP-25 are given in Table 4*S-
TABLE 4.8
Pour
Kinematic
Viscosity,
est
Lubricating Capacity
on Four- Ball
Tester
(dba = 19 mil)
Extent of Corrosion
in Pinkevich Device
(at a Temperature of + 70*'C
for 50 hrs)
Point,
at
4-100" C
at
-35* C
Critical
Load Per,
kg
mdth
of Wear
^ot^ mm
Steel
SOKhGSA
Alloy
BrAZhMts
10-3-1.5
Brass
LS-59
^56
10.2
23660
64
0.85
+0.11
4-0,26
4-0,24
As the test results indicate, the oil WII NP-25 is close to the MS-20 oil
itfith respect to lubricatirig properties during vibration.
Bearings of all types working on the oil VNII NP-25 show little wear both
at positive and negative tenperatures.
An Mprovement in the lubricating properties of oils may constitute an inir-
portant factor promoting a pronounced increase in the lifetime of main and tail
rotor hubs of helicopters. Therefore, studies in this direction will acquire an
1^0
ever greater scope. In such investigations, consideration should be given to /395
the at)Ove-descri*bed characteristics of the mechanism of wear and lubrication con-
ditions of highly loaded vibrating bearings -
Oils for needle bearings of flapping and drag hinges . These bearings, as
a rule, are less loaded than the bearings of feathering hinges, so that they are
not as sensitive to the properties of the lubricant. The selection of lubri-
cating materials for needle bearings of the flapping and drag hinges of rotor
hubs is facilitated by the fact that solidification of the lubricant when the
rotor is inoperative leads to no unfavorable consequences. In the flapping and
drag hinges (if they are present) of tail rotors, the lubricant cannot be per-
mitted to solidify since increased moments of friction in these assemblies may
result in shaking of the helicopter.
At present, "hypoid" lubricants are used in the flapping and drag hinges of
the main and tail rotor hubs of Soviet helicopters. Practical e:xperience with
helicopter operation has shown that hypoid oils, despite their content of free
sulfur at ordinary specific pressures, ensure a sufficiently long life for vi-
brating needle bearings. Ifypoid lubricants, just as other oils with antiwear
additives, are unsuitable for feathering hinges.
Fig.4-51 Feathering Hinge of Rotor Hub.
Ifypoid oil has high tackiness and hence provides the necessary lubrication
for contacting elements even if the hubs are not conpletely tight. The replace-
ment of hypoid oil by greases (which is sometimes resorted to in tail rotors of
helicopters operating at especially low tenperatures) greatly shortens the
service life of needle bearings in flapping and drag hinges.
3. Calculation of Hub Bearings in Main and Tail Rotors
Bearings of feathering hinges * Figure 4 -SI shows a typical design of
feathering hinges for the main rotor hubs of Soviet helicopters.
In calculations of bearings for the feathering hinges of rotor hubs, it is
connnon to take into account the centrifugal force of the blade N and the moment
in the plane of rotation M^ created by the damper.
A41
In feathering hinges manufactured according to the scheme shown in Eig#4»5l*
the centrif-ugal force of the "blade is alDsorlDed Iby the thrust bearing (!)• The
moment of the datrper is alDsorbed in part "by the same bearing and in part bgr the
radial bearings (2) and (3).
The loads on the radial bearings (2) and (3) in flight are conparatively
small; conseqaently, they are -usually selected from static considerations based
on the weight moment of the blade transmitted to them when the helicopter is /396
standing, the rotor is not rotating, and the blades abut the coning stops ♦ As
shown in practical use, the loads on the radial bearings of feathering hinges
due to the weight moment of the blade may go as high as 100 - 110^ of their
static load capacity catalog rating.
J)isplacem€nt
of cage
zsS
cm
t
\-\
21000
\
\
K
18000
'^
,__j
—
L^
\kQOO
-^
Q
S'iO'*
tO'W^
' 15 W'
^ IQ'W
' nh^
'fir
Pig*4»52 Thrust Bearing with Slewed
Rollers; Recording of Motion of the
Bearing Gage during Vibration.
Fig.Z|.»53 Curve of life Expectancy
a = a(nh) for Thrust Bearings with
Slewed Rollers.
The life e3q)ectancy of thrust bearings of feathering hinges is calculated
on the basis of the experimental relation a = a(nh) obtained from tests with the
proper types of bearings under vibration conditions at purely axial load. For
thrust ball bearings, the curve of life expectancy a = a(nh) is plotted in
Fig .4.49 •
As mentioned before, thrust bearings with slewed rollers are being used in
the feathering hinges of rotor hubs of all series-produced Soviet helicopters.
The basic diagram. of such bearings is shown in Pig .4* 52 • Thanks to arrangement
of the seats of the cage at an angle to the radial direction in bearings of this
type, the cage not only vibrates together with the revolving race but also shifts
continuoiisly, although very slowly, in the same direction. This continuous dis-
placement of the cage prevents "brinelling" of the race tracks and leads to a
substantial increase of the load-carrying capacity of the bearing.
Tests have established that the life e^qjectancy of thrust bearings with
slewed rollers largely depends on the rate of displacement of the retainer. This
rate is commonly characterized by the time T^, during which the cage turns
through an angle of 360°. The optimal values of the time T^ for the vibration
ajiplitudes and frequencies at which the thrust bearings of feathering hinges
operate are 40-80 min. When T^ > 80 min, the probability of failure of the
bearing due to spallLng of the metal on the rollers increases. Despite continu-
A42
ous displacement of the cage, the same surface areas of the rollers are in con-
tact with the races • Therefore, failure of thrust bearings with slewed rollers
begins in most cases with damage to the rollers. It should be mentioned that,
at T(5 = 2*5 to 6 hrs, the durability of the rollers drops by a factor of about 2.
When Tg < 40 min, friction losses and wear of the tracks increase noticeably*
The curve of life expectancy a = a(nh) for thrust bearings with slewed rol-
lers having an optimal radial displacement of the cage of T^ = 40 - 80 min is
shown in Fig .4* 53* This curve has been plotted from test results with several
batches of such bearings and MS-20 oil at an anplitude of the revolving race /397
cpo = 4*5° and a frequency n = 240 cyc/min, i.e., under conditions analogous to
the test conditions whose results were used in constructing the relation a =
= a(nh) in RLg.4«49«
Bench tests and operating experience indicate that the curves of Hfe ex-
pectancy plotted in Figs .4*49 and 4*53 can be used for determining the rated
service life of rotor hub thrust bearings for all operating conditions these units
encounter under real conditions.
As we see from Figs .4*49 and 4*53, the equation of the life expectancy
curves a = a(nh) for vibrating bearings has the same foi*m as for bearings rotating
in one direction:
a'"'
(nh)=^QonsU (5.1)
where we have m^^" = 10 for the case of point contact and m^^ = 6.66 for the case of
linear contact.
Let us take for the base the product nh = 120,000, which corresponds ap-
proxljnately to a 500- hour operating life of helicopters of the Mi-1 type. At
nh = 120,000 the permissible contact stresses are 29^000 kg/cm^ for thrust ball
bearings and 18,800 kg/cm^ for thrust bearings with slewed rollers. Let Aq de-
note the axial force which, in a bearing with uniform distribution of forces
over the rolling bodies, sets up contact stresses equal to those permissible at
nh = 120,000. Then, in conformity with eq.(3«l) the permissible force on the
ball will be
p _ ^q/ 120 000 \Q.3
^'-'7{-lJr) (5-2)
Here we have taken into account that for ball bearings the contact stresses
are proportional to the cube root and for roller bearings, to the square root
of the load.
Special experiments have established that the moment which must be taken
into account in calculating the service life of a thriist bearing for feathering
hinges is about 25 - 50^ of the moment of the dainper, depending on the design
featiores of the assembly and on the clearances. Here the calculation is per-
formed in terms of the instantaneous maximum pressure on the rolling body,
meaning that the moment acting on the thrust bearing of the feathering hinge is
arbitrarily considered as constant in magnitude and direction.
A43
The XQBjdjmm. pressure on the rolling body of a thrust bearing, loaded by an
axial force and moment, can be represented in the form
z
Comparing this equality with eq.(5«2), we obtain the following expression
for determining the rated service Hfe of thrust bearings of feathering hinges
for rotor hubs:
/ nh \Q.3 _ ^0
\i2000oy "" 4^>// ' (5*3)
^ follows from Sections 3 and 4, the coefficient ko'^^ depends on the rela-
tive eccentricity of application of the axial force, which in this case is
equal to
T = (0,25 -^0.5)^. , ,,
''-0^ (5.4)
For the usual correlations between the moment of the danper and the /395
centrifugal force, t does not exceed 0.1; therefore, all rolling elements are
always loaded in the thrust bearings of feathering hinges'"".
For thrust roller bearings in which all rollers share the load, we have
kW^\-^2i. (5.5)
For small values of t, the coefficients k^^^ for thrust ball and roller
bearings practically coincide. This permits use of eq.(5.5) even for calcula-
tions of thrust ball bearings.
From eqs.(5.4) and (5*5), we finally find
«M, = i+ (0.5-1)^. (5.6)
It should be noted that the calculation of radial-thrust bearings of dif-
ferent types intended for service in feathering hinges of main and tail rotor
hubs can also be performed by means of eqs.(5.3) and (5*5) if "the permissible
axial loads Aq corresponding to the value nh = 120,000 are predetermined for
these bearings. Here it is assumed that the moments acting on the bearings are
known from calculations or experiments.
It must be remembered that the values of Aq which were not obtained from
''^ It is known that the relative eccentricity at which unloaded rolling elements
occur is 0.5 for thrust roller bearings and 0.6 for thrust ball bearings.
444
the coirplete ciirve of life expectancy cr = cr(nh) relative to a certain pro"babil-
ity of failure of the bearings but from a recoirputation based on results of ex-
periments carried out at some value of nh for an insufficiently large number of
test specimens, may be incorrect; apparently, this is associated with an ap-
preciable dispersion of the life expectancy, which is difficult to elicit at a
single load level.
Fig.4^54 Feathering Hinge of Rotor Hub, on Multi-Row
Radial-Thrust Bearing.
Multi-row radial-thrust ball bearings with contact angles of Po =45° and
a reduced ratio of track radius to ball diameter are being successfully used in
the feathering hinges of main and tail rotor hubs of certain helicopters (see
Fig«Zf54)« This ratio is usually equal to 0.515 in antifriction bearings. In 7399
the mentioned multi-row bearings it has been reduced to 0.510, which leads to a
decrease in contact stresses by about 1% and thus to an increase in the rated
service life of the bearings by a factor of 2. It is logical that such a way of
increasing the load-carrying capacity of radial-thrust ball bearings is useful
mainly for the case of vibrations, since a reduction in the ratio of track radi-
us to ball diameter increases the length of the area of contact strain due to
which the friction losses increase noticeably. Test results indicate that, in
the case of high-quality manufacture ensuring a sufficiently uniform distribu-
tion of the external load over the bearings of the assembly, the permissible
contact stresses on whose basis the axial force must be calculated are here
24,000 kg/cm^ for multi-row radial-thrust ball bearings.
Available data on the permissible contact stresses in radial-thrust roller
bearings for service under vibration conditions are still insufficiently veri-
fied.
The above values of permissible contact stresses pertain to cases of the
service of feathering hinges in main and tail rotor hubs with oils not inferior
in lubricating properties to the oils MS- 20 and MS-14* If this requirement is
not met, these values must be reduced accordingly.
445
The permissible contact stresses are noticea"bly affected Iby the size of
the rolling elements so that, when using large bearings^ a certain correction
should be introduced for the scale factor. As shown by test results, the values
of the permissible stresses given above can be considered binding for bearings
with balls up to 25 nmi in diameter and rollers up to 15 mm in diameter. On
changing from rollers with a diameter of 15 mm to rollers with a diameter of
24 mm, the permissible contact stresses for thrust bearings with slewed rollers
drop by about 10% •
Needle Bearings of Flapping and Drag Hinges * In most rotor hubs, needle
bearings are used for the flapping and drag hinges.
The performance of needle bearings is usually estimated in terms of the
magnitude of specific pressure per unit area of projection of the track of the
inner race.
In calculations of needle bearings for drag hinges it is generally assumed
that the load is uniformly distributed over the length of the needles (see
Fig.4*55^a), In conformity with this, the specific pressure for bearings is
taken as equal to
DU ' (5.7)
where
D = diameter of the track of the inner race;
t£ = total working length of the needles.
Needle bearings for flapping hinges, in addition to the centrifi:igal force
of the blade N, take a certain moment M (Fig •4.55, b) whose constant component M^
is determined with sufficient accuracy by the expression
^-S-Ht;- (5.a)
Here,
Mpot = torque of the rotor;
Zpot ~ number of blades of the rotor;
a = "drift" of the middle of the flapping hinge from the axis of /400
rotation;
Is = distance between flapping and vertical hinges;
t^^h = "offset" of drag hinge.
The variable component M^ of the moment M, when calculating needle bearings
of the flapping hinges of rotor hubs, is disregarded since it has little effect
on their life expectancy. It is customary to assime that the load in flapping
hinges manufactured in conformity with the scheme in Fig . 4-55, b is distributed
over the length of the bearings according to the trapezoidal rule. In this case,
the loaded state of the bearings is characterized by the specific pressures qi
and q^ on the outer edges of the races caused by the combined action of the
446
force N and the moment M^^ These pressures are calculated ty means of the for-
mula
7>.=^±6.
Ma
DB2
(-1)'
(5.9)
where B is the working width of the "bearing assembly.
Substituting into eq«(5»9) the value of Ma, we reduce it to the form
q -^
1
~{'-^J
(5.10)
As design data indicate, proper choice of the "drift »' a permits approach-
ing the specific pressures q^ and q^ sufficiently close to the average specific
pressure qo =
N
Dl,
in the basic powered flight regijnes . We note that the
"drift" of the middle of the flapping hinge from the axis of rotation by a dis-
tance a is equivalent to rotation of this hinge through an angle ^^^^ =
tan
r-l
(see Fig .4. 55) •
'V, h
Fig .4 • 55 For Calculation of Needle Bearings of Flapping
and Drag Hinges of Rotor Hubs.
According to eq.(5*10), the specific pressures qi and qs depend on the cenr-
trifugal force N and on the torque Mj.ot • Therefore, these can be regarded as /kOl.
certain functions of the rotor ipm and power. After using eq.(5.lO) for plot-
A47
ill
ting the graphs of q^ = qiCN^.^^) and q^ and qsC^j.^^ ) for the most characteristic
rotor ipm, as is done in Pig .4 •56, it "becomes easy to determine the values of
the specific pressiires qx and q^ in the main flight regimes of a helicopter and
also to estimate the correctness of selec-
tion of the "drift" a and, if necessary,
to introduce 'suitable corrections into the
rotor hut) design.
If the flapping hinges are made in the
form of two independent sipports whose spac-
ing L substantially exceeds the diameter of
the track D (Fig .4*57) ^ it can be assumed
that, within each support, the specific
pressures in the bearings are constant.
In this case, the rated specific pres-
sures determining the life expectancy of the
N" N' N^'^ *"' needle bearings in the flapping hinges are
''''* ^° '^° equal to
Fig .4*56 Specific Pressures qi
and qg as a Function of Rotor
Rpm and Power.
ri^ot 9 ^rot = rotor rpm and power
in cruising regime; njot^ ^iot =
= rotor rpm and power at cruis-
ing speed; n^J? , ^^1% = rotor
x*pm and power in takeoff regime;
n^S't, N,"ot = rotor rpm and
power in autorotation regime.
Fig .4.57 For Calculation of
Widely Spaced Needle Bearings
of Flapping Hinges.
Ql,2 =
N
(5.11)
where t^ is the total length of the needles
in both bearings .
When using hypoid lubricants, the per-
missible specific pressures in well-sealed
needle bearings corresponding to a life ex-
pectancy of 1000 hrs at 240 cyc/min are at
least 350 kg/cm^ for the flapping hinges and
400 kg/cm^ for the drag hinges. The rela-
tively small value of the permissible speci-
fic pressures in flapping hinge bearings
can be e^q^lained in part by the fact that
they work at vibration airplitudes of 2 to 6*^,
whereas the vibration amplitude of the drag
hinge bearings usually does not exceed 1°.
Although this contradicts established opin-
ions, practical use has shown that, at vi-
bration anplitudes to 1^, the life expect-
ancy of needle bearings is higher than at
anplitudes of 2 - 6°. The fact that, due to
deformation of the parts under load, the actual specific pressures on the edges
of the needle bearings of flapping hinges may at times exceed the rated pres-
sures possibly plays a definite role here.
Many years of actual service experience confirm that, in selecting the /402
448
size of needle bearings for the flapping and drag hinges of rotor hubs of light
and mediiom helicopters, the above-indicated values of permissible specific pres-
siores can be used as a reliable g\rLde. For heavy helicopters whose units gen-
erally have a relatively lower rigidity, these figures can be used as guide only
Fig .4.59 Failure of Needle Bearing
due to Insufficient Rigidity of
the Structure.
d^^
Fig .4.5s Effect of Stiffness
of Pin and Flexibility of Races
on the Distribution of Specific
Pressures over the Length of
Needle Bearings in a Flapping
Hinge,
a - Initial version; b - Effect
of pin of increased stiffness;
c - Effect of "flexible" ends
of bearing races .
M^Mn^-M
u »
Fig .4.60 For Calculation
of Needle Bearings in
Tail Rotor Hubs.
if special measures are taken to ensure a uniform load distribution in the drag
hinge bearings and if the diagram of the load distribution in the flapping
hinges approximates a trapezoidal diagram (see Fig.4»555t)) • As a rule, a satis-
factory load distribution over the length of needle bearings for flapping and
drag hinges can be obtained by properly choosing the stiffness of rings and pins
and also by suitably raising the flexibility of the ends of the races. This is
shown specifically in Fig.4»58 which contains experimental diagrams of the vari-
ation in distance between the generatrices of the outer and inner races, for
A49
three design versions of the flapping hinge in the rotor hub of a heavy heli-
copter. It should te noted that inadequate mechanical strength of the rings and
pins in flapping and drag hinges may not only result in local increases of the
depth of "brinelling marks at the edges of the tracks "but also in spalUng of
large portions of their surface and sometimes even in "breakage of the needles
(Rig.4.59).
Calculation of needle bearings for flapping hinges of tail rotor hubs /Z[.03
(i^g*4*60) is appreciably more difficult than calculation of needle bearings for
flapping hinges of main rotor hubs, since they generally absorb a rather large
alternating moment which cannot be disregarded in estimating their performance.
This moment is created by alternating aerodynamic and inertia (Coriolis) forces
acting on the blades of the tail rotor in the plane of rotation. In rough cal-
culations, the loaded state of needle bearings in tail rotor flapping hinges is
usually characterized by the instantaneous maximum specific pressure set ijp on
the edge of the track. On the assimption that the load is distributed over the
length of the bearings in accordance "with the trapezoidal rule, this pressure
is equal to
A^
^+M. ^
i-j^e-^t. ^
BN
(-^r
(5.12)
where
Mt,r = torque of tail rotor;
Zt.r = blade number of tail rotor;
M^ = ajrplitude of variable moment loading the flapping hinge.
The values of the specific pressure q calculated from eq.(5*l2) for tail
rotors of light and medium helicopters at cruising speed should not exceed
300 - 350 kg/cm^. When hypoid lubricants are used in the flapping hinges, it
can be expected that the life expectancy of the bearings will be at least
1000 hrs.
Finally, the life expectancy of needle bearings for flapping and drag
hinges of main and tail rotor hubs is determined from tests of such units on
special rigs.
4. Calculation of Bearings for the Pitch Control and
Control Mechanisms
The permissible loads on the bearings of the pitch-control hinges and their
connecting control elements generally are determined by experiment. For this,
endurance tests are performed on special rather coirplex installations which per-
mit simulating all types of forces acting on the pitch control in flight.
The loads on the pitch control are of a dynamic nature. This is especially
clear from the oscillograph in Fig .4*61, for the blade hinge moment M^ and the
forces Piong ^^^ Plat ^^ "^h® longitudinal and lateral control rods connecting
450
the corresponding rockers of the pitch control with the h^^draullc boosters <
It is logical that, with such a coirplex character of loading, any recom^
mendations as to the design of bearings for pitch control hinges vdll of neces-
sity be only conditional. Nevertheless, certain suggestions might guide the
designer in problems of the selection of bearings for these vital units; in this
respect, we will briefly discuss these.
Mu
/hok
lon^ ^\
'lal n
y\f\r\r\
AVN/W^
Fig. 4 -61 Oscillograms for Blade
Hinge Moment and Forces in
Longitudinal and lateral
Control Rods .
Fig.Z4..62 Load on Bearings
of Pitch Control Hinges.
If we take into consideration that, in conventional rotor designs, only the
absolute magnitude of the blade hinge moment changes and that the correlation
between the anplitudes and phases of its individual harmonics remains constant,
then the selection of bearings for such hinges of the pitch control based on the
same design configuration can proceed from the maximum value of the absorbed load
P. ax (Fig, 4. 62).
For pitch controls close in design to the pitch controls of Mi-1 and Mi-4
helicopters (see KLg.4.38) with all-metal rotor blades of rectangular planform
and using greases of the type TsIATIM-201, the permissible load ^%l\^ can be de-
termined from Table 4*9 • This table was coirpiled from results of stand tests
with consideration of practical experience in operating pitch controls.
The values of the permissible loads Pperm given in the Table for a rotor /4Q5
rpm of 240 correspond to a life expectancy of 1000 - 1200 hrs . For other rpm,
the Hfe expectancy is found from the expression
/z^
240,000
(5.13)
where n is the rated rotor rpm.
451
TABI^E 4.9
Site
of
Installation
Hinges of swashplate,
turn rod, and levers
of blade
Bearings of universal
joint
Bearings of rockers
of longitudinal and
lateral controls
Elearings of longitudinal
and lateral control rods
connecting the rockers
with outer race of Cardan
joint
Bearings of collective
pitch lever
Permissible Values ?"■'' (kg) for
\t • T* - perm '-'
Various Bearings
Ball,
Radial,
Radial -
Thrust,
and Thrust
0.8 Q^
Qst
Ball.
Spherical
Qst
Roller,
Spherical
0.8 Qst
Roller,
Radial-
Thrust,
and Thrust
Qst
Qst
0.8 Qrf
0-8 Qst
Needle
2DI
2DI
Hinge
of Type
ShS
2DI
Db
Qst ~ permissible static load on a nonrotating bearing, given in
catalogs and manuals;
D = diameter of inner race track of needle bearing or spheres
of hinged bearing, in mm;
b = width of outer race of hinged bearing, in mm;
I - working length of the needles, in mm.
If the nature of the loads differs from that of the pitch controls of the
Mi-1 and Mi-4 helicopters with all-metal rotor blades, then the permissible
values Pperm should be refined as a result of appropriate stand and service
tests*
Above, we have examined vibrating bearings that execute a large number of
vibrations (more than lO"^) during the rated service life.
The permissible loads on the bearings of the control mechanism of aircraft,
for which the total number of vibrations does not exceed 100,000 and the vibra-
tion anplitude is equal to 20^ and more, should be determined - according to
TOIPP - by the following experimental formula"'^:
/?,
perm
"perm
Zd].
(5-lff)
The values of the coefficient ol
perm
for certain types of bearings operating
on greases at vibration nijmbers 25,000 and 100,000 are given in Table 4*10.
"" It is assumed that the contact areas of adjacent rolling elements do not over-
lap.
452
TABLE 4*10
Type
of
Bearing
Designation
of Bearing
Inside
Diaineter
of Bearing,
mm
Value of Coefficient o^^
At 25000
Vibrations
At 100,000
Vibrations
Ball, radial
Ball, spherical
7000100
100
200
900000
980000
981000
1000
1200
1300
971000
to 50
to 9
abovi 9
to 9
above 9
to 9
above 9
to 10
tohO
tohO
2.5
2
4.7
2
1.6
2
1.6
2
1.6
2.8
2.8
3.3
Section 6. Theory_and Selection _of _Basic Parameters of Thrust
Bear ings with "Slewed" Rollers
7406
As indicated in previous Sections, thrust hearings with cylindrical rollers
arranged at an angle to the radial direction are being used with success in the
feathering hinges of rotor hubs of Soviet helicopters. The high load-carrying
capacity of such bearings, known as thrust bearings with "slewed" rollers, is ex-
plained by the fact that the cage, during vibrations, not only vibrates together
with the revolving race but also shifts continuously in one direction. The time
of rotation of the cage T^ through an angle of 360°, characterizing the rate of
this displacement, is determined by a number of factors. It is dependent on
the coefficient of sUding friction between roller and races, vibration airpll-
tude and frequency of the revolving race, and on a number of geometric parame-
ters of which the angles of slope of the cage seats play a major role. It is
logical that these angles should be selected such that the time Tc will be with-
in optimal limits ensuring a long life expectancy of the rollers at acceptable
wear of the tracks. A theory is presented below by means of which this problem
can be solved.
453
1. Determination of the Time T^
In thrust hearings with slewed rollers, the ratio of angular velocity of
the cage to angular velocity of the revolving race A =
tu.
CD
depends vpon the
direction of rotation. This causes a continuous displacement of the cage, which
is Cbserved in such bearings during viTDration.
The values of the ratio A corresponding to counterclockwise and clockwise
rotation of the hearing are found in the following manner:
The forces of sliding friction arising at the points of contact of the
roller with the races are reduced to the resultant forces F^y, F^^ , Fgy , F^^,
and the moments M-l^ , Mg^ (Fig.4*63)« At a constant coefficient of sHding fric-
tion |JL between rollers and races, the magnitudes of these forces and moments can
he calculated with sufficient accuracy by the formulas
(6-1)
Here,
P = force absorbed by the roller in question;
y^ and y^ = coordinates of the contact points at which there is no sHding
in a direction perpendicular to the roller axis;
dj, = diameter of the roller;
t = working length of the roller.
In deriAd^ng eqs.(6.l), it was assumed that the normal loads qi and qg / lj.07
are distributed over the roller length according to the law
where
P 21
(6.2)
454
Such a distribution of normal loads is due to the action of the moment
(^ly "*' ^sy) — ^-- which tends to turn the roller about the axis Oj,x« Since the
usual load concentration at the edges of the roller has little effect on the
time Tq, we will disregard it to simpUfy the calculations. In view of the
smallness of the friction force ij.c(Fix - Fsx) "we consider that
The coefficients entering eqs«(6.1) are deteinnined 'by the equalities
Mobile
race
V , , ^i ^
'///m//^C
X ■A^k^'^'z
B
( 1 1 \l/2 •
IT + ^J
- P 1/1 IV
5u = 125,0 (^-5.o);
1 \l/2'
'-'11 — >
C -^"
1-13- —
(6.3)
Fig. 4.63 Forces and Moments
Acting on "Slewed" Rollers.
where
/"O sin Y
From the kinematic relations and the equations of moments relative to /kO^
the axis Oj.7, we can obtain the following expressions'" :
„ _ i ± ^'^ il ^'o
(2A
-1);
(lil^c)
/ An
. ' ^" J\
(6.4)
* It is assiimed that the quantity ^,o(2A - 1) can "be neglected for lonity. The
inertia moment of the roller is disregarded.
455
112 ± 2/
+
(iTl^c)M^^^^
(2.A-1).
In these expressions.
y Jro
M^
Pd^ '
where
fj.o = coefficient of rolling friction;
Mj. = moment taking into account the friction at the ends of the roller
and the friction against the lulbricant;
jjLc - coefficient of friction "between the rollers and cage*
The upper signs refer to the case of F^^ " ^sx ^ 0, while the lower signs
indicate the case of Fi^ - f'sx ^ 0.
The angle of slope y is considered positive if the roller can be placed in
a radial position "by turning about the point 0^ counterclockwise. Under this
condition, the directions indicated in Fig .4*63 correspond to positive values
of the forces calculated by eqs.(6.l). The signs of the angles of slope of . the
rollers and the direction of rotation are determined when viewing the roller
from the side of the movable race.
A roller with an angle of slope y generates the following moment, relative
to the axis of rotation of the cage:
As shown by calculations, the resistance to rolling and the friction of the
roller against the cage and lubricant have practically no effect on the magni-
tude of this moment. In conformity with this, taking f = |jIq = into account
and considering that in real designs the angle is y < 6*^ and hence cos y ^ 1,
the last equality, by means of eqs.(6.l), (6.3), and (6-4)> can be transformed
such that
A/=rfxPro/W,
(6.5)
where
XT o ^
M
2t^|f-^io-2^nf-(2A-l).
Table 4*11 gives the values of the coefficients A^^ and An as a function
of the quantity l/p#
456
TABIE 4.U.
AQ2
VbIucb
of the
Coeffi*
cienta
AlQ Mid
All •'
Vp
0.05
0.87702
1,99008
0.10
1.28664
1.96117
0.15
1.50479
1.91565
0.20
0.25
0.30
0.35
0.40
0.45
0,50
2
1.60798
1.85695
1.63952
1.78885
1.62687
1.71498
1.58789
1.63846
1.53409
1.66173
1.47287
1.48659
1.40889
1.41421
In the case of negative values of p, the coefficients A^q and A^^^ can be
determined "by means of the relation
^lo^- P)= ~ ^xoiP)
and
(6.6)
These relations follow directly from eqs.(6.3)»
At small vibration amplitudes, when the inertia forces can be disregarded,
the equation of motion of the cage of the bearing with "slewed" rollers reduces
to the condition
M,l-Mfr=0,
(6.7)
Here,
M
g^^ = ^ = total moment of the sUding friction forces exerted on the rol-
lers by the bearing races;
Mfj. = moment of friction.
Let us assume that the cage has z seats, in each of which are s rollers.
The angles of slope of the cage seats, at an average radius re are denoted in
terms of y^ , and the angles of slope of the rollers in terms of Yij^ • The sub-
script i denotes the number of the cage seat, while the subscript k indicates
the position of the roller in it. Usually, in each seat there are two rollers.
The load on a roller with a working length t^ is equal to
(6.8)
where
l^ = total working length of the rollers in one seat;
N = axial force appHed to the bearing.
A change in direction of rotation of the bearing is equivalent to a change
in signs of the angles of slope of the rollers. Bearing in mind this fact, we
obtain the following expression from eqs.(6.5), (6.6), and (6.S):
457
t S
■(2A-1)
where A^^q (pjj^ ) and A^^Cpij^ ) are the values of the coefficients A^q and A^ for
For definitiveness, we will consider that the signs of the angles of /U2D
slope of the rollers are given for the case of counterclockwise rotation of the
bearing. After suTDstituting eq.(6.9) into eq.(6.7), we obtain;
for counterclockwise rotation :
1 ^'i . Mfr
A^A'=-1-+-1 '-'*-' =
2 2 z s -2
(6.10)
for clockwise rotation :
1 ^Iz Mfr
A = A" = -i-
1 ^ i^lk^t
11. ^ ^
(6.11)
/_1 Jfe«l
Knowing the quantities A' and k" , it is easy to calculate the time T^ . From
Pig#4»52 it follows that, during each half-period of vi"brations, the cage is
displaced iDy an angle Acp^ = (A' - A''')cpo* Consequently, the time of rotation of
the cage through an angle of 360° will l)e
360
21 A' -A"|yon
where n = is the num^ber of vilDrations of the revolving race per minute.
To
Since the moment of friction Mfj. of the cage should "be independent of the
direction of rotation, we will have, in conformity with the alDOve correlations,
458
r,=
180 2rc /-I ftTi '^'=
Von(* rf^
SStt^'-w
f-i ft-i
(6.12)
TABLE 4.12
Time Tg (min) at Oil Temperature
+.(20-30) —(30-40) —(45-55)
63
54
68
Equation (6.12) is the main formula for
the theory of a thrust "bearing -with a "slewed"
roller. It follows specifically that friction
of the cage does not affect the time T^. This
important conclusion is confirmed "by results
of experiments set vp to determine the time T^
at low tenperatures, when, owing to an increase
in viscosity of the oil, the moment M^j. may at-
tain an appreciable magnitude (Table 4.12)*
Oil VNII NP-25 (v = 10 centistokes
at t = +100OC and V = 50,000 centi-
stokes at t = -40°C)
2. Selection of Angles of Slope
of Cage Seats
Ml
A prescribed rate of displacement of the
cage is ensured by proper selection of the
angles of slope of its seats. In this case, not only the rated values of the
angles but also the allowances for man-ufacture, which have a noticeable effect
on the time T^, must be kept in mind. The remaining geometric -parameters of the
bearing, influencing the time T^
tions.
Fig .4 -64 Determination of the
Values of p with Clearance be-
tween Rollers and Cage.
are selected on the basis of design considera-
Manufacturing deviations of the angles
of slope of the cage seats, even with up-
to-date technology and rigorous quality con-
trol of the finished articles, go as high as
7 - 10'. If no special measures are taken
in the manufacture of bearings, such devia-
tions may reach 20 - 30'.
The time T^ depends also on the clear-
ance between rollers and cage. In the pres-
ence of clearance, the position of the rol-
lers in the seats of the cage and hence the
actual angles of slope of the rollers are
determined by the forces of sliding friction
exerted on the rollers by the races. Since,
in the general case, a determination of
these forces is difficult, we will assume
that the rollers, with equal probability,
can occupy any of two positions shown in
Fig .4 •64 s
in position I:
459
^c sin Vc.H- — /-c sin ^Yc. + —j
in position II:
„. — 11^. ^^^.
It is obvious from these equalities that the effect of clearance on the
time Tc can be taken into account by increasing the design deviations of the
angles of slope of the cage seats to the quantity
2/-. ■
' C
where ^m is half of the manufactizring allowance, while e„ax is the maximum
clearance.
The most general case of arrangement of the cage seats of practical inter-
est is the case where the cage contains Zi seats with an angle of slope Yi ±
± 5(Yi ^ O) ^^d 22 seats with an angle of slope ± 5»
To simplify further calculations, let us assume that i ^-^ = I = const. /412
If the quantity 5 is such that the difference A' - A '^ is positive, then, pro-
vided that ^j^ = 1 = const, the time T^ can change from a certain
y.(max) ^ 180 2/-C ^u{pl)+X^n{ P2)
cpo^V dr ^10(/';)+X^10(7j (6.13)
to a certain
Here,
y,(mln) ^ 180 2rc ^U (p]) + XA^^ (p^)
fonvix" d^ Ao{p.])+xAio{p"2y' (6.14)
Pl=-~zi:~7r-< p:= '
rcSin(Yi — $) ' 2 Ac sin S
^ /■cSin(Yi + £) ' ^ /-cSinS
s^
460
The coefficient v is usually close to unity (v = 0.9S - l) . This indicates
that the time T^ depends little on the numtier of rollers s in one seat.
In the expressions for T^^^^^ and T^^^^^ , the minimum and maximum values
of the coefficient of sliding friction ijl are denoted "by |jb' and p,''. With good
lubrication, we have ji' ?^ 0.05 and p.'' ;^ 0.08.
Let the upper and lower limits of the range of optimal values of T^, be
equal to T' and T^' respectively. As follows from test results, thrust bearings
with slewed rollers operating in the feathering hinges of rotor hubs will have
a time T^ = 80 min and T^' = 40 min. It has been established that, to determine
the maximum stability of the rate of displacement of the cage, the quantities Vx
and X should be selected such that, at a given value of 5, T^'^^^'^will be equal
to T^. At T^"'^''^ = T^, the quantities Yi> X, and ? are correlated by a definite
relation. Setting T^''^''^ = T^ in eq.(6.13), this relation can be ^graphically
represented as a fand-ly of corresponding curves'"". Figure 4*65 which gives a
family of such curves shows that the condition T^^^^"^^ = T^ imposes certain limi-
Thus in the case T' ==
15',
tations on the selection of the quantities y i and 5
SO min, the ratio x should not exceed 1.28 for 5=0 and 0.77_ for
and the angle y^ should not be less than some minimiom angle Vi
(min) _
+ 5
(where Y^ is the value of y^^^""^ for x = and 5=0). The range of time rate
of change of cage displacement is characterized by the ratio T\ - T^^min) ^r^Ua x)
12 X
Fig .4.65 Curves of y 1 == Yi(x) fo^
Different Values of the Deviation
of Z.
n
^"]
n
0.6
0.5
OA
0.3
i
\
[
^
-JO".
- -
0,1
OJ
-
—
I .
.._
.,- ^ ~
M2.
10^ 20' I
Fig .4 •66 Dependence of the Ratio T]
on the Deviation of 5 for
Different Angles of y ^ .
Figure 4.66 contains the curves of T] =_T|(|) for the angles yx = 5° and
Y^ = 30', plotted on the assuirption that T^"''^
the ratio T] depends mainly on the quantity g
effect on Tl. Thus, from the viewpoint of stability of the rate of cage displace-
ment, different combinations of the angles of slope of the seats are approxi-
mately equivalent, provided they satisfy the conditions T^e'"'"'' = T^ . According
^(max) ^ rp/^ Figtire 4-66 shows that
The angle y^ has a rather minor
""" Everywhere where no special stipulations have been made, it is assiimed that
T^ = 80 min. Here, all specific numerical values pertain to the case dj. = 9 mm,
re = 40 mm, t = 8 mm, v = 1, cpo = 4»5°, n = 240 cyc/min, jj, = O.O6.
461
to Fig.4^66, the deviation of § at which T^^^^^ = T^' = 40 min and thus T] =
= JiQ« = o#5, is about 5'. Hence, even with the most careful manufacture of
80
cages, there might "be cases in which the time T^, will exceed the limits of the
optimal range.
In practice, we encounter two variants of arranging the cage seats. In the
first, all seats have an identical angle of slope not exceeding 1° while in the
second, several seats are arranged at an angle of 3 - 6° with all other seats
being radial. Let us conpare these variants for the following exanples.
Let us examine a "bearing for which Yi == 45 ', X = 0, dj. = 9 imii, r^ = 40 mm,
t = 8 mm, e^ax = 0*2 rm, ^^ = 7', and s = 2; the hearing operates at cpo - 4*5°
and n = 240 cjc/rain.
If all rollers have an angle of slope equal to y^ we have
J. ^ 180 2re Au(p)
where
r^sixi y
Figure 4-6? gives the curves of T^, = T^Cy), showing the variation in the
time Tc as a function of the angle y ^or |ji = p.' = 0.05 and [i = \x^^ =^ O.OS. We
distinguish between the curves of T^, = Tc(y) a region bounded bj vertical
straight lines Y = Yx + ?m + -^^^^^ = 60' and y = Yi - 5m - ^"'"'^ == 30' . The
2rc 2re
actual values of the time T^ should lie within this region. It is easy to note
that, for such a bearing, T^^^^""^ = 74 min and T^^'"'^ = 31 min. These values are
rather close to optimal. Results of experiments set up to determine the time T^
for several hundreds of bearings with the indicated parameters have shown that
the actual values of T^. for all practical purposes do not extend beyond the
limits of the indicated range, being groiped about average values of T^^^^ =
=50-60 min.
Now let the bearing have the following parameters: Yi = 5^, X = 5, dj. =
= 5 mm, re = 28 mm, I = 4*2 mm, e^^ax = 0.2 mm, 5^ = 7', s = 2 and let it oper-
ate at cpo = 4*5^ and n = 300 cyc/min.
¥e assume that the actual angles of slope of the radial seats are equal /414
to 5 . Then,
<Ponif.v dr\Aio(Pi)~hxA^oiP2)\ ' (6.16)
where
462
Pi^^ — : ^^^ P2^
re sin Yi r^sin 8
Flgiare ^•6B which gives the curves of T^ = T^C^) plotted from eq.(6.l6)
shows that, at § =0, the tme is T^ = I63 - 261 min depending on the friction
coefficient iJ. . If 5 ?« -5', then the time T^ will tend to infinity. In other
words, at small negative deviations of the angles of slope of the radial seats,
the cage may stop moving. Such cases are often observed when testing bearings
with large values of x •
r^ min
WO
Fig .4 -67 The Time T^ as a Func-
tion of the Angle of Slope of
the Gage Seats Y •
JO' r^
Fig.4»68 The Time Tc as a Function
of the Deviation of 5 .
At iJi = 0.08 and ? = 5^ **■
2re
= 15', we have T^ = 47 min. Consequently,
in the case in question the time T^ may vary from T^"i^^^ = co to T^"'^^^ = 47 min.
These exanples indicate that only the first variant of positioning the cage
seats enables the bearings to operate under conditions close to optimal.
Positioning of the seats of the cage at identical angles will also reduce
the friction losses and the nonuniformity of distribution of the normal load
along the contact lines.
3 . Friction Losses
Friction losses in thrust bearings with "slewed" rollers depend both on the
463
rate of the displacement of the cage and on the mode of selection of the quanti-
ties Yi ^^ X which provide cage displacement at a given rate.
The moment of friction of the iDearing is usually written as /415
^/.--^A^r,, (6 .17)
where ffj. is the reduced friction coefficient-
Using the relations olDtained in the preceding subsections, after a number
of transformations we find
ffr-fro + f,&^' (6-18)
Here,
fj.o = coefficient of rolUng friction;
f st = coefficient characterizing losses due to sliding friction.
The coefficient f e i can be represented in the form [see (Ref.27)]
f,\=^
^^Ju(P±
Tc Aio(pj) _^-
T-f* * ^11 (P^)
(6.19)
I c *— — .
On hand of Table 4»10 it is easy to demonstrate that, with an increase in
angle of slope of the rollers, the ratio — ' — -^ , ' first increases rapidly,
Aii(p)
reaching, at y ~ sin""^ 0.57 ^ /^c 9 ^ value equal to unity. Upon a further in-
crease in the angle, the ratio — -r — will not change. This means that the
Aii(p)
quantity T^^^ represents the minimum time obtainable at a given value of the
friction coefficient y. [see eq.(6.15)].
Figure 4*69 indicates that, at the same rate of cage displacement, the fric-
tion losses decrease with decreasing angle Yi* Hence, the minimum friction
losses actually occur when all cage seats are positioned at identical angles to
the radial direction. At optimal rates of cage displacement f ^ O.5V
464
non-o*bservance of this arrangement of seats may lead to an Increase in sliding
friction losses loy a factor of 1.5*
4» Additional Considerat ions of Optim al Thrust Bearing Design
TO.th "Slewed" Rollers
According to the a"bove formulas, the coefficient K characterizing the non-
uniformity of load distrilDution over the roller length, is equal to
K=6^'-^{2B,o^^^B,,J^^^y
Since | 2Bio | ^ Im- — ~ B
curacy that K = l2iJi>
dr
M-
dp
21
- Bio
yi - Js
, we can consider with sufficient ac-
21
Thus, the coefficient K depends only on the
angle y ^it follows from the curve of K = K(y) plotted in Fig •4*70 that, on
changing from an angle of 5° to an angle of 45' which corresponds to the posi-
tioning of all seats at identical angles, the coefficient K will decrease from
0.35 to 0.14-
The arrangement of all cage seats at equal angles is preferatile also in /416
view of the following considerations: If the angle of slope of the seats is
identical, the forces F^ - Fg^ driving the rollers against their lateral
surfaces are very small. If Mf^. = and s = 1, these forces are theoretically
absent.
a 2 OM 0.6 0,8 10 ^
0.3
0,1
0.1
.,_-■■ — X-^ —
Fig .4 -69 Losses Due to Sliding
Friction.
Fig.4»70 Coefficient K as a Func-
tion of the Roller Angle of Slope.
At different angles of slope of the seats, when the "slewed" rollers must
overcome the resistance of radially arranged rollers, the forces F^^ - Fg may
attain substantial magnitudes (tp to 0.1 (j,P) and cause wear of the cage (espe-
cially at large x) •
465
So far, it has been assumed that all rollers are of identical length. Now
let us see what happens at an alternation of long and short rollers in staggered
sequence.
Table 4»13 shows that, in the latter case, the time T^ and the reduced co-
efficient of friction ff j. vary negligibly, whereas the coefficient K for shoii:
rollers increases by a factor of 3* This indicates that it is expedient to use
rollers of the same length in thriist bearings with slewed rollers.
TABLE 4.13
EFFECT OF DISTRIBUTION OF ROLLER LMGTH ON T^ , ff^ , AND K
Variant of Distribution
of Roller Lengths
7-c
min
//r (when
4-0.003)
K
Rollers of identical length
(/l=/2=8 mm )
Long and short rollers alternating
in staggered sequence
(/j = n mm , [2=5 mm )
48
45.7
0.00616
0.00674
0.14
0.43
(for short
rollers)
In estimating the effect of a nonuniform load distribution caused by the
/ . dj.
action of the moment (F^^ + F^ ; on the life e:xpectancy of a bearing, one
must not lose sight of the fact that the load at each contact point does not re-
main constant but changes with any change in direction of rotation* In particu-
lar, at the ends of the rollers the normal load varies in accordance with the
law
.-f±ii'^if
As a consequence, the nonuniformity of load distribution caused by the /417
effect of the above moment should not excessively reduce the service life of the
bearing.
The usual concentration of load on the ends of the rollers, which we have
disregarded assi:iming that q = const at y = 0^ is of ijnportance. To lessen the
detrimental effect of the latter, it is preferable to use rollers with a camber.
5 . Fbcample of Calculating; a Thrust Bearing with "Slewed" Rollers
In conclusion, let us give an exanple of calculating a thrust bearing with
"slewed" rollers.
466
Given: axial load N = 20,000 kg, vibration anplitude of revolving race 90 =
= 4«5*^, frequency n = 180 cyc/min.
For the given conditions, we select a "bearing with the following parame-
ters: dp = 12 mm, r^ = 61 mm, I = 10.5 nim (total length of rollers: I ' = 12 mm),
z = 20, and s = 2.
Wanted: to determine the angles of slope of the cage seats that will en-
sure optimal rate of displacement and maximimi service life of the "bearings.
We calculate the coefficient v:
^.'- 1 1
v= ^^^ „ = 1 = 1 =^0.99.
^ /-^ / /' \2 / 12 \2
Let us assume that all seats have an identical angle of slope. After sub-
stituting into eq.(6.15) cpo == 4*5°, n = IBO cyc/min, v = 0.99, and iJ, = jx' =
= 0.05, we will construct the ciorve of T^ = T^Cy) by means of Table 4*11* From
the curve we find the value y' of the angle y ^"t which T^, = T^ = 80 min. In
our case, y' = kh' • Taking 5^1 = 7' and e^jax = 0.18 mm, we determine the devia-
tion of 5 :
5^7 + 57.3x60^:^=12'.
' 2x61
The normal value of the angles of slope of the seats of the cage is
Y='Y'+?=46+12 = 58^
The contact stresses in the bearing are
a = 860i/^^=860i / ?5^522 ;:=; 17000 kq)cm .
According to Fig .4*53, nh = 27 ^ 10^ corresponds to this value of a. Con-
sequently, the service life of the bearing is
fl= -=1500 hri.
180
467
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470
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